ci.  0 lo 


THE  ETHEL  CARR  PEACOCK 
MEMORIAL  COLLECTION 

Matris  amori  monumentum 


TRINITY  COLLEGE  LIBRARY 


DURHAM,  N.  C. 

1903 

Gift  of  Dr.  and  Mrs.  Dred  Peacock 


Digitized  by  the  Internet  Archive 
in  2016  with  funding  from 
Duke  University  Libraries 


https://archive.org/details/mathematics21  hutt 


A 


COURSE 

OF 


MATHEMATICS. 

At IN  TWO  VOLUMES. 


FOR  THE  USE  OF  ACADEMIES, 

es@J/r&4 


AS  WELL  AS 


PRIVATE  TUITION. 


if 


BY 

CHARLES  HUTTON,  L.L.D.  F.R.S. 

LATE  PROFESSOR  OF  MATHEMATICS  IN  THE  ROYAL  MILITARY 
ACADEMY. 


FROM  TEE  FIFTH  AND  SIXTH  LONDON  EDITIONS? 

REVISED  AND  CORRECTED  BY 

ROBERT  ADRAIN,  A.  M. 

FELLOW  OF  THE  AMERICAN  PHILOSOPHICAL  SOCIETY, 

AND 

PROFESSOR  OF  MATHEMATICS  IN  QUEEN’S  COLLEGE,  NEW-3ERSEY, 


VOL.  II. 


NEW-YORK  : 

PUBLISHED  BY  SAMUEL  CAMPBELL,  EVERT  DUYCKINCK, 
T.  & J.  SWORDS,  PETER  A.  MESIER,  R.  M’DERMUT, 
TII03LV.S  A.  RONALDS,  JOHN  TEBBOUT, 

AND  GEORGE  LONG. 


1818. 

ZJH-Tl 


4 


VV'(  * •;>, 


George  Long,  Printer. 


i H 


d~/o 

H 

P 

CONTENTS 

OF  VOLUME  U. 


PLANE  TRIGONOMETRY  considered  analytically 

Spherical  Trigonometry 

On  Geodesic  Operations,  and  the  Figure  of  the  Earth 

Principles  of  Polygonometry 

Of  Motion,  Forces 

General  Laws  of  Motion 

Collision  of  Radies  ....... 


Laws  of  Gravity— Descent  of  heavy  bodies — Motion  of  Projectiles  in 
free  space 
Practical  Gunnery 

Descent  of  bodies  on  inclined  planes  and  curve  surfaces — Moti 
pendulums  .... 

The  Mechanical  Powers 
Centre  of  Gravity  . 

Strength  and  stress  oj  beams,  &c. 

Centre  of  Percussion 
Centre  oj  Oscillation 
Centre  of  Gyration 
Of  Hydrostatics 


Of  Hydraulics 
Of  Pneumatics 
Of  the  Syphon 
Of  the  Air-Pump 
Diving-bell  and  condensing  machine 
Barometer  ■ ...  . 


Page 

1 

5 

59 

96 

109 

111 

121 


128 
141 

on  of 

144 
154 
169 
181 

195 

196 
199 
201 
212 
217 
227 

229 

230 
232 

Thermometer 


X 3 ^ V | 


VI 


CONTENTS. 


V 

v-  - \ 

l>age  , 

Thermometer 233 

On  the  resistance  of  Fluids , and  -with  their  forces  and  actions  on 

bodies 236 

Practical  Exercises  concerning  Specific  Gravity  . . . 239 

Of  the  Piling  of  Balls  and  Shells  ......  245 

OJ  Distances  by  the  velocity  of  Sound  247 

Practical  Exercises  in  Mechanics,  Statics,  Hydraulics , Sound,  Mo- 
tion, Gravity,  Projectiles,  and  other  branches  oj  Natural  Philo* 

sophy . 243 

On  the  Nature  and  Solution  of  Equations  in  general  , . 25! 

On  the  Nature  and  Properties  of  Curves , and  the  Construction  of  Geo- 
metrical Problems 273 

THE  DOCTRINE  OF  FLUXIONS. 

Definitions  and  Principles 304 

Direct  Method  of  Fluxions  ...  .....  30S 

Of  second,  third,  idc.  Fluxions 314 

The  Inverse  Method,  or  The  Finding  of  Fluents  ....  319 

Of  Fluxions  and  Fluents 333 

Of  Maxima  and  Minima  ........  351 

The  Method  of  Tangents 356 

Of  Rectifications,  or,  to  find  the  lengths  of  Curve  Lines  . . 358 

Of  Quadratures 360 

To  find  the  Surfaces  of  Solids . 362 

To  find  the  Contents  oj  Solids  ' . 363 

To  find  Logarithms 364 

To  find  the  points  of  Infection 366 

To  find  the  radius  of  curvature  of  Curves  .....  366 

Of  Involute  and  Evolute  Curves  . . . . . . . . 370 

To  find  (he  Centre  of  Gravity . ST S 

Practical 


I 


CONTENTS. 


rii 

Page 

Practical  Questions 375 

Practical  Exercises  concerning  Forces  . . . . . 378 

On  the  Motion  of  Bodies  in  Fluids 402 

On  the  Motion  of  Machines,  and  their  Maximum  effects  . . 4]  6 

Pressure  oj  Earth  and  Fluids  against  -malls  and  Fortifications— 

Theory  of  Magazines,  &c.  432 

Theory  and  Practice  of  Gunnery 444 

Promiscuous  Problems  and  Exercises  in  Mechanics,  Statics,  Dynam- 
[ ics,  Hydrostatics,  Hydraulics,  Projectiles,  &c.  Sdc.  . . . 480 

Additions 555 

Tables  of  logarithms,  Sines,  and  Tangents-  ....  559 


1 


' . ■-  ■ ■ 


% 


.• 


► 


. 


4 


ji 


COURSE 

• p 

OF 

MATHEMATICS,  &c. 


PLANE  TRIGONOMETRY  CONSIDERED  ANALYTICALLY, 


Art.  1. 

HERE  are  two  methods  which  are  adopted  by  mathemati- 
cians in  investigating  the  theory  of  Trigonometry  : the  one 
Geometrical,  the  other  Algebraical.  In  the  former,  the  vari- 
ous relations  of  the  sines,  cosines,  tangents,  &c.  of  single  or 
multiple  arcs  or  angles,  and  those  of  the  sides  and  angles  of 
triangles,  are  deduced  immediately  from  the  figures  to  which 
the  several  enquiries  are  referred  ; each  individual  case  requir- 
ing its  own  particular  method,  and  resting  on  evidence  peculi- 
ar to  itself.  In  the  latter,  the  nature  and  properties  of  the  lin- 
ear-angular quantities  (sines,  tangents,  &c.)  being  first  defined, 
some  general  relation  of  these  quantities,  or  of  them  in  con- 
nection with  a triangle,  is  expressed  by  one  or  more  algebra- 
ical equations  ; and  then  every  other  theorem  or  precept,  of 
use  in  this  branch  of  science,  is  developed  by  the  simple  re- 
duction and  transformation  of  the  primitive  equation.  Thus, 
the  rules  for  the  three  fundamental  cases  in  Plane  Trigonome- 
try, which  are  deduced  by  three  independent  geometrical  in- 
vestigations, in  the  first  volume  of  this  Course  of  Mathematics, 
are  obtained  algebraically,  by  forming,  between  the  three  data 
Ve*.  II.  ,®  and 


ANALYTICAL  PLANE  TRIGONOMETRY. 


2 

and  the  three  unknown  quantities,  three  equations,  and  obtain- 
ing, in  expressions  of  known  terms,  the  value  of  each  of  the  un- 
known quantities,  the  others  being  exterminated  by  the  usual 
processes.  Each  of  these  general  methods  has  its  peculiar  ad- 
vantages. The  geometrical  method  carries  conviction  at  every 
step  ; and  by  keeping  the  objects  of  enquiry  constantly  before 
the  eye  of  the  student,  serves  admirably  to  guard  him  against 
the  admission  of  error  : the  algebraical  method,  on  the  con- 
trary, requiring  little  aid  from  first  principles,  but  merely  at 
the  commencement  of  its  career,  is  more  properly  mechanical 
than  mental,  and  requires  frequent  checks  to  prevent  any 
deviation  from  truth.  The  geometrical  method  is  direct, 
and  rapid  in  producing  the  requisite  conclusions  at  the  out- 
set of  trigonometrical  science  ; but  slow  and  circuitous  in 
arriving  at  those  results  which  the  modern  state  of  the  science 
requires  : while  the  algebraical  method,  though  -sometimes 
circuitous  in  the  developement  of  the  mere  elementary  theo- 
rems, is  very  rapid  and  fertile  in  producing  those  curious  and 
interesting  formulae,  which  are  wanted  in  the  higher  branches 
of  pure  analysis,  and  in  mixed  mathematics,  especially  in 
Physical  Astronomy.  This  mode  of  developing  the  theory 
of  Trigonometry  is,  consequently,  well  suited  for  the  use  of 
the  more  advanced  student  ; and  is  therefore  introduced  here 
with  as  much  brevity  as  is  consistent  with  its  nature  and 
utility. 

2.  To  save  the  trouble  of  turning  very  frequently  to  the 
1st  volume,  a few  of  the  principal  definitions,  there  given, 
are  here  repeated,  as  follows  : 

The  sine  of  an  arc  is  the  perpendicular  let  fall  from  one 
of  its  extremities  upon  the  diameter  of  the  circle  which 
passes  through  the  other  extremity. 

The  cosine  of  an  arc,  is  the  sine  of  the  complement  of 
that  arc,  and  is  equal  to  the  part  of  the  radius  comprised  be- 
tween the  centre  of  the  circle  and  the  foot  of  the  sine. 

The  tangent  of  an  arc,  is  a line  which  touches  the  circle 
in  one  extremity  of  that  arc,  and  is  continued  from  thence 
till  it  meets  a line  drawn  from  or  through  the  centre  and 
through  the  other  extremity  of  the  arc. 

The  secant  of  an  arc,  is  the  radius  drawn  through  one 
of  the  extremities  of  that  arc  and  prolonged  till  it  meets  the 
tangent  drawn  from  the  other  extremity. 

The  versed  sine  of  an  arc,  is  that  part  of  the  diameter 
of  the  circle  which  lies  between  the  beginning  of  the  arc  and 
the  foot  of  the  sine. 

The  cotangent,  cosecant,  and  coversed  sine  of  an 
arc,  are  the  taDgent,  secant,  and  versed  sine,  of  the  comple- 
ment of  such  arc.  3.  Since 


ANALYTICAL  PLANE  TRIGONOMETRY. 


O 


-3.  Since  arcs  are  proper  and  adequate  measures  of  plane 
angles,  (the  ratio  of  any  two  plane  angles  being  constantly 
.equal  to  the  ratio  of  the  two  arcs  of  any  circle  whose  centre 
is  the  angular  point,  and  which  are  intercepted  by  the  lines 
whose  inclinations  form  the  angle),  it  is  usual,  and  it  is  per- 
fectly safe,  to  apply  the  above  names  without  circumlocution 
as  though  they  referred  to  the  angles  themselves  ; thus,  when 
we  speak  of  the  sine,  tangent,  or  secant,  of  an  angle,  we 
mean  the  sine,  tangent,  or  secant,  of  the  arc  which  measures 
that  angle  ; the  radius  of  the  circle  employed  being  known. 

4.  It  has  been  shown  in  the  1st  vol.  (pa.  382),  that  the  tan- 
gent is  a fourth  proportional  to  the  cosine,  sine,  and  radius  ; 
the  secant,  a third  proportional  to  the  cosine  and  radius  ; the 
cotangent,  a fourth  proportional  to  the  sine,  cosine,  and  ra- 
dius ; and  the  cosecant  a third  proportional  to  the  sine  and 
radius.  Hence,  making  use  of  the  obvious  abbreviations, 
and  converting  the  analogies  into  equations,  we  have 

rad.  x sine.  rad.  x cos.  rad.  . rad2. 

fan.  = , cot.  , sec.  = , cosec. = . 

cos.  sine  cos-  sine 

Or,  assuming  unity  for  the  rad.  of  the  circle,  these  will  become 
sin  cos,  1 1 

tan.  = — ...  cot.  — — ...  sec.  = — ...  cosec.  = — . 
cos.  sin.  cos.  sin. 

These  preliminaries  being  borne  in  mind,  the  student  may 
pursue  his  investigations. 

5.  Let  abc  be  any  plane  triangle,  of 
which  the  side  bc  opposite  the  angle  a is 
denoted  by  the  small  letter  a,  the  side  ac 
opposite  the  angle  b by  the  small  letter  b, 
and  the  side  ab  opposite  the  angle  c by 
the  small  letter  c,  and  ci>  perpendicular  to  ab  ; then  is, 
c = a . cos  b -j-  b . cos  a. 

For,  since  ac  = b,  ad  is  the  cosine  of  a to  that  radius  ; 
consequently,  supposing  radius  to  be  unity,  we  have  ad  =-b, 
cos.  a.  In  like  manner  it  is  bd  = a . cos.  e.  Therefore, 

ad  + bd  = ab  = c — a . cos.  b -f-  b . cos.  a.  By  pursuing 

similar  reasoning  with  respect  to  the  other  two  sides  of  the 
triangle  exactly  analogous  results  will  be  obtained.  Placed 
together,  they  will  be  as  below  : 

a=  b . cos.  c-fc.  cos.  b. 
b = a . cos.  c + c . cos.  a. 

c = a . cos.  b + b . cos.  a. 

6.  Now,  if  from  these  equations  it  were  required  to  find 
expressions  for  the  angles  of  a plane  triangle,  when  the  sides 

are 


4 


ANALYTICAL  PLANE  TRIGONOMETRY. 


are  given  ; we  have  only  to  multiply  the  first  of  these  equa- 
tions by  a,  the  second  by  b,  the  third  by  c,  aod  to  subtract 
each  of  the  equations  thus  obtained  from  the  sum  of  the  other 
two.  For  thus  we  shall  have 

b 2 4-  c2  - a 2 ' 

h?  4*  c — or  — 2bc  . cos.  a,  whence  cos.  a = 


a2  -j-  c" — b 2 = 2ae  . cos.  b, 


a + l2 — c2  = 2 ab  . cos,  c, 


2 t>c 

a 2 -p  i2  — b2 


COS.  B,  = 

2 ac 

a2+b * _ c* 

COS.  C,  = 

2 an 


(II.) 


7.  More  convenient  expressions  than  these  will  be  deduc- 
ed hereafter  : but  even  these  will  often  be  found  very  con- 
venient, when  the  sides  of  triangles  are  expressed  in  integers, 
and  tables  of  sines  and  tangents,  as  well  as  a table  of  squares; 
(like  that  in  our  first  vol.)  are  at  hand. 

Suppose,  for  example,  the  sides  of  the  triangle  are  a = 320, 
7)  = 562,  c = 800,  being  the  numbers  given  in  prop.  4,  pa. 
161,  of  the  Introduction  to  the  Mathematical  Tables  : then 
we  have 

b2+c2  __a2  = 853444  log.  = 5-9311751 

2 be  . . — 899200  log.  = 5-9638080 

The  remainder  being  log.  cos.  a,  or  of  18°20'  = 9-9773671 
Again,  a2  + c2  — b2  = 426556  . . . log.  = 5-6299760 

2ac  . . . = 512000  . . . log.  = 5-7092700 

The  remainder  being  log.  cos.  b,  or  of  33°35'  = 9-9207060 

Then  180°—  (18°  20'+33°  35')  = 128°  5'  = cTTwherl  all 
the  three  triangles  are  determined  in  7 lines. 

8.  If  it  were  wished  to  get  expressions  for  the  sines,  in- 
stead of  the  cosines,  of  the  angles  ; it  would  merely  be  ne- 
cessary to  introduce  into  the  preceding  equations  (marked  II), 
instead  of  cos.  a,  cos.  b,  &c.  their  equivalents  cos.  a = ^/  (1  — 
sin2,  a),  cos.  b = ^ (1  — sin2,  b),  &c.  For  then,  after  a little 
reduction,  there  would  result, 


sin.  a = 2a2 b2  + 2a2  c2  -{-  262c2  — (a4-f-64+c4) 

sin.  b = ^7  y/  2a2  b2  + 2a2 c2  -f  262c2  - (a4+64-f  c4)  - 

sin.  c = 5ab  \/  2 a2b2  + 2a2  c2  + 2 b2c2  — (a4+64+c4) 

- 

Or,  resolving  the  expression  under  the  radical  into  its  four 
constituent  factors,  substituting  s for  a -f-  b -j-  c,  and  reducing, 
.the  equations  will  become. 

Sin. 


ANALYTICAL  PLANE  TRIGONOMETRY. 


o 


sin.  a = fc  As  (As  — a)  (|s  — b)  (As  — c)  1 

sin.  b = s y AS  (as  — a)  (As— 6)  (|s  — c)  j.  (III.) 

sin.  c = £ v/  is  (as  — a)  (as  — b ) (As  — c)  j 

These  equations  are  moderately  well  suited  for  computation 
in  their  latter  form  ; they  are  also  perfectly  symmetrical  ; and 
as  indeed  the  quantities  under  the  radical  are  identical,  and  are 
constituted  of  known  terms,  they  may  be  represented  by  the 
same  character  ; suppose  k : then  shall  we  have 

. 2k  . 2k  . 2k  ..... 

sin.  a = — - . . . sin.  b = — ...  sin.  c = — ...  (m.) 

be  ac  au 

Hence  we  may  immediately  deduce  a very  important  theo- 

rem : for,  the  first  of  these  equations,  divided  by  the  second 

gives  and  the  first  divided  by  the  third  gives 

sin.  b b 

sin.  a a . . 

= — : whence  we  have 

sin.  c c 

sin.  a : sin.  b . sin.  c a a : b : c ..  . (IV.) 

Or,  in  words,  the  sides  of  plane  triangles'  are  proportional  to 
the  sines  of  their  opposite  angles.  (See  th.  1,  Trig.  vol.  i). 

9.  Before  the  remainder  of  the  theorems,  necessary  in  the 
solution  of  plane  triangles,  are  investigated,  the  fundamental 
proposition  in  the  theory  of  sines,  &c.  must  be  deduced,  and 
the  method  explained  by  which  Tables  of  these  quantities, 
confined  within  the  limits  of  the  quadrant,  are  made  to  ex- 
tend to  the  whole  circle,  or  to  any  number  of  quadrants 
whatever.  In  order  to  this,  expressions  must  be  first  ob- 
tained for  the  sines,  cosine,  &c.  of  the  sums  and  differences 
of  any  two  arcs  or  angles.  Now,  it  has  been  found  (I)  that 
a = b . cos.  c + c . cos.  b.  And  the  equations  (IV)  give 
» sin.  b sin.  c _ . . . , 

o—a.  . . . . c=a.  . substituting  these  values 

sm.  a sin.  a 

of  b and  c for  them  in  the  preceding  equation,  and  multiplying 
the  whole  by  — > it  will  become 

sin.  a = sin.  b . cos.  c -f-  sin.  c . cos.  b. 

But,  in  every  plane  triangle,  the  sum  of  the  three  angles  is 
two  right  angles  ; therefore  b and  c are  equal  to  the  supple- 
ment of  a : and,  consequently,  since  an  angle  and  its  suppler 
ment  have  the  same  sine  (cor.  1,  pa.  378,  vol.  i),  we  have  sin. 
(b  4*  c)  = sin,  b . cos.  c + sin.  c,  cos  . e. 


10.  If, 


6 


ANALYTICAL  PLANE  TRIGONOMETRY, 


10.  If,  in  the  last  equation,  c become  subtractive,  then 
would  sin.  c manifestly  become  subtractive  also,  while  the 
cosine  of  c would  not  change  its  sign,  since  it  wrould  still  con- 
tinue to  be  estimated  on  the  same  radius  in  the  same  direction. 
Hence  the  preceding  equation  would  become. 

sin.  (b  — c)  = sin.  b . cos.  c — sin.  c . cos.  b. 

11.  Let  c be  the  complement  of  c,  and  be  the  quarter 

of  the  circumference  : then  will  c'  ={  o — c.  sin.  c'  = cos.  c, 
and  cos.  c'  = sin.  c.  But  (art.  10),  sin.  (b  — o')  = sin.  b, 
cos.  c'  — sin.  c'  cos.  b.  Therefore,  substituting  for  sin.  c', 
cos  c',  their  values,  there  will  result  sin.  (b  — c')  = sin.  b . 
sin.  c.  — cos.  b . cos.  c.  but  because  c'  = — c>  we  have 

sin.  (b  — c)  = sin.  (b  + c — |o)  = sin.  [(b  + c)  — i O ] = 
— sin.  [|0  — (b  + c)]  = — cos.  (b  -j-  c).  Substituting  this 
value  of  sin.  (b  — c')  in  the  equation  above,  it  becomes  cos. 
(b  -f-  c)  = cos,  b . cos.  c.  — sin.  b . sin.  c. 

12.  In  this  latter  equation,  if  c be  made  subtractive,  sin.  c 
will  become  — sin.  c,  while  cos.  c will  not  change  : conse- 
quently the  equation  will  be  transformed  to  the  following, 
viz.  cos.  (b  — c)  = cos.  b.  cos.  c -f-  sin.  b.  sin.  c. 

If,  instead  of  the  angles  b and  c,  the  angles  had  been  a and 
b ; or,  if  a and  b represented  the  arcs  which  measure  those 
angles,  the  results  would  evidently  be  similar  : they  may  there- 
fore be  expressed  generally  by  the  two  following  equations, 
for  the  sines  and  cosines  of  the  sums  or  differences  of  any  two 
arcs  or  angles  : 

sin.  (a  ± b)  = sin.  a . cos.  b.  ± sin.  b . cos.  a. 

cos.  (a  ± b)  = cos.  a . cos.  b.  zp  sin.  a . sin.  b 

13.  We  are  now  in  a state  to  trace  completely  the  muta- 
tions of  the  sines,  cosines,  &c.  as  they  relate  to  arcs  in  the  va- 
rious parts  of  a circle  ; and  thence  to  perceive  that  tables 
which  apparently  are  included  within  a quadrant,  are,  in  fact, 
applicable  to  the  whole  circle. 

Imagine  that  the  radius  mc  of  the  circle,  in  the  marginal 
figure,  coinciding  at  first  with  ac,  turns  about  the  point  c (in 
the  same  manner  as  a rod  would  turn  on  a pivot),  and  thus 

forming  successively  with  ac  all 
possible  angles  : the  point  m at 
its  extremity  passing  over  all 
the  points  of  the  circumference 
aba'b'a,  or  describing  the  whole 
circle.  Tracing  this  motion  at- 
tentively, it  will  appear,  that  at 
the  point  a,  where  the  arc  is 
nothing,  the  sine  is  nothing  also, 
while  the  cosine  does  not  differ 


from 


analytical  plane  trigonometry.  7 

from  the  radius.  As  the  radius  mc  recedes  from  ac,  the  sine 
pm  keeps  increasing,  and  the  cosine  cp  decreasing,  till  the 
describing  point  m has  passed  over  a quadrant,  and  arrived 
at  b : in  that  case  pm  becomes  equal  to  cb  the  radius,  and 
the  cosine  cp  vanishes.  The  point  m continuing  its  motion 
beyond  b,  the  sine  p'm'  will  diminish,  while  the  cosine  cp', 
which  now  falls  on  the  contrary  side  of  the  centre  c will  in- 
crease. In  the  figure,  p' m'  and  cp'  are  respectively  the  sine 
and  cosine  of  the  arc  iV  or  the  sine  and  cosine  of  abm', 
which  is  the  supplement  of  a'm'  to  half  the  circumfe- 
rence : whence  it  follows  that  an  obtuce  angle  (measured  by 
an  arc  greater  than  a quadrant)  has  the  same  sine  and  cosine 
as  its  supplement  ; the  cosine  however,  being  reckoned  sub- 
tractive or  negative,  because  it  is  situated  contrariwise  with 
regard  to  the  centre  c. 

When  the  describing  point  m has  passed  over  a or  hall 
the  circumference,  and  has  arrived  at  a',  the  sine  p'm'  va- 
nishes, or  becomes  nothing,  as  at  the  point  a,  and  the  cosine 
is  again  equal  to  the  radius  of  the  circle.  Here  the  angle 
acm  has  attained  its  maximum  limit  ; but  the  radius  cm  may 
still  be  supposed  to  continue  its  motion,  and  pass  below  the 
diameter  A.  The  sine,  which  will  then  be  p"m",  will  con- 
sequently fall  below  the  diameter,  and  will  augment  as  it 
moves  along  the  third  quadrant,  while  on  the  contrary  cp", 
the  cosine,  will  diminish.  In  this  quadrant  too,  both  sine 
and  cosine  must  be  considered  as  negative  : the  former  being 
en  a contrary  side  of  the  diameter,  the  latter  a contrary  side 
of  the  centre,  to  what  each  was  respectively  in  the  first  quad- 
rant. At  the  point  b',  where  the  arc  is  three-fourths  of  the 
circumference,  or  fO,  the  sine  p"  u"  becomes  equal  to  the 
radius  cb,  and  the  cosine  cp"  vanishes.  Finally,  in  the  fourth 
quadrant,  from  b'  to  a,  the  sine  p'"u'",  always  below  aa di- 
minishes in  its  progress,  while  the  cosine  cp'",  which  is  then 
found  on  the  same  side  of  the  centre  as  it  was  in  the  first 
quadrant,  augments  till  it  becomes  equal  to  the  radius  ca. 
Hence,  the  sine  in  this  quadrant  is  to  be  considered  as  nega- 
tive or  subtractive,  the  cosine  as  positive.  If  the  motion  of 
m were  continued  through  the  circumference  again,  the  cir- 
cumstances would  be  exactly  the  same  in  the  fifth  quadrant  as 
in  the  first,  in  the  sixth  as  in  the  second,  in  the  seventh  as  in. 
the  third,  in  the  eighth  as  in  the  fourth  : and  the  like  would 
be  the  case  in  any  subsequent  revolutions. 

14.  If  the  mutations  of  the  tangent  be  traced  in  like  man- 
ner, it  will  be  seen  that  its  magnitude  passes  from  nothing  to 
infinity  in  the  first  quadrant  ; becomes  negative,  and  de- 
creases from  infinity  to  nothing  in  the  second  ; becomes  po- 
sitive again,  and  increases  from  nothing  to  infinity  in  the 

third 


8 


ANALYTICAL  FLANE  TRIG0NGMETRY. 


third  quadrant ; and  lastly,  becomes  negative  again,  and  de 
creases  from  infinity  to  nothing,  in  the  fourth  quadrant. 

15.  These  conclusions  admit  of  a ready  confirmation  ; and 
others  may  be  deduced,  by  means  of  the  analytical  expres- 
sions in  arts.  4 and  12.  Thus,  if  a be  supposed  equal  to  JO> 
in  equa.  v,  it  will  become. 

cos.  (JO  ± b)  = cos.  J O-  cos.  B ^ sin-  | O • sin.  b, 

sin.  (JQ  — B)  = sin-  I O-  cos.  B — s*n-  B • cos.  J o 

But  sin.  { O = rad.  = 1 ; and  cos.  J Q = 0 : 
so  that  the  above  equations  will  become 
cos.  ({  O ± B)  = + sin.  b. 

sin.  (i  O — B)  = cos.  b. 

From  which  it  is  obvious,  that  if  the  sine  and  cosine  of  a u 
arc,  less  than  a quadrant,  be  regarded  as  positive,  the  cosine 
of  an  arc  greater  than  JO  aQd  less  than  J Q will  be  negative, 
but  its  sine  positive  If  b also  be  made  = J o 5 then  shall 
we  have  cos.  JO  = — 1 ; sin  J O = 0. 

Suppose  next,  that  in  the  equa.  v,  a = J o 5 then  shall  we 
obtain. 

cos.  (|  O ± b = — cos.  B. 
sin.  (J  O ± B = ± sin.  e ; 

which  indicates,  that  every  arc  comprised  between  J Q and 
|Q,  or  that  terminates  in  the  third  quadrant,  will  have  its 
sine  and  its  cosine  both  negative.  In  this  case  too,  when 
b =J  0>  or  the  arc  terminates  at  the  end  of  the  third  quad- 
rant, we  shall  have  coss  J O = 0,  sin.  | O = — 1. 

Lastly  the  case  remains  to  be  considered  in  which  a =J  O 
or  in  which  the  arc  terminates  in  the  fourth  quadrant.  Here 
the  primitive  equations  (V)  give 

cos.  (|  o ± b)  = ± sin.  b. 
sin.  (|  O ± B)  = — cos.  b ; 

so  that  in  all  arcs  between  £ O and  Q.  the  cosines  are  posi- 
tive and  the  sines  negative. 

16.  The  changes  of  the  tangents,  with  regard  to  positive 
and  negative,  may  be  traced  by  the  application  of  the  pre- 
ceding results  to  the  algebraic  expression  for  the  tangent  ; viz. 

sin. 

tan.  = . For  it  is  hence  manifest,  that  when  the  sine  and 

cos. 

cosine  are  either  both  positive  or  both  negative,  the  tangent 
will  be  positive  ; which  will  be  the  case  in  the  first  and  third 
quadrants.  But  when  the  sine  and  cosine  have  different 
signs,  the  tangents  will  be  negative,  as  in  the  second  and 
fourth  quadrants.  The  algebraic  expression  for  the  cotan- 
cos. 

gent,  viz.  cot.  = , will  produce  exactly  the  same  results. 

sin. 


The 


.ANALYTICAL  PLANE  TRIGONOMETRY.  Q 


The  expressions  for  the  secants  and  cosecants,  viz.  sec.  = 
1 l 

■ — , cosec.  = — show,  that  the  signs  of  the  secants  are  the 

cos.  s:n. 

same  as  those  of  the  cosines  ; and  those  of  the  cosecants  the 
same  as  those  of  the  sines. 

The  magnitude  of  the  tangent  at  the  end  of  the  first  and 
third  quadrants  will  be  infinite  ; because  in  those  places  the 
sine  is  equal  to  radius,  the  cosine  equal  to  zero,  and  therefore 

— co  (infinity).  Of  these,  however,  the  former  will  be 

reckoned  positive,  the  latter  negative. 

17  The  magnitudes  of  the  cotangents,  secants,-  and  cose- 
cants may  be  tr  ced  in  like  manner  ; and  the  results  of  the 
13th,  14th,  and  15th  articles,  recapitulated  and  tabulated  as 
below. 


0° 

90° 

180° 

270° 

360° 'l 

Sin. 

0 

R 

0 

— R 

0 

Tan. 

0 

oo 

0 

— CO 

0 

Sec. 

u 

CO 

R 

— CO 

R 

Cos. 

R 

0 

— R 

0 

R 

Cot. 

CO 

0 

CO 

0 

CO 

Cosec. 

co 

R 

— co 

— R 

CO  j 

The  changes  of  signs  are  these. 


(VI.) 


1st.  5th.  9th.  13th. 
2d.  6th.  10th.  14th 
3d.  7th.  11th.  15th. 
4th.  8th.  12th.  16th. 


sin.  cos.  tan.  cot.  sec.  cosec< 

: + + + + + + ) 

±Z  + + Z±((vno 

J- 

We  have  been  thus  particular  in  tracing  the  mutations, 
both  with  regard  to  value  and  algebraic  signs,  of  the  princi- 
pal trigonometrical  quantities,  because  ^.knowledge  of  them, 
is  absolutely  necessary  in  the  application  of  trigonometry  to 
the  solution  of  equations,  and  to  various  astronomical  and 
physical  problems. 

18.  We  may  now  proceed  to  the  investigation  of  other  ex- 
pressions relating  to  the  sums,  differences,  multiples,  &c.  of 
arcs  ; and  in  order  that  these  expressions  may  have  the  more 
generality,  give  to  the  radius  any  value  r instead  of  confining 
it  to  unity.  This  indeed  may  always  be  done  in  an  expres- 
sion, however  complex,  by  merely  rendering  all  the  terms 
homogeneous  ; that  is,  by  multiplying  each  term  by  such  a, 
power  of  r as  shall  make,  it  of  the  same  dimension,  as  the 
term  in  the  equation  which  has  the  highest  dimension . Thus, 
the  expression  for  a triple  arc. 

II:  3 sin. 


10 


ANALYTICAL  PLANE  TRIGONOMETRY. 


sin.  3a  = 3 sin.  a — 4 sin3,  a (radius  = 1) 
becomes  when  radius  is  assumed  = R, 

R2  sin.  3a  = r2  3 sin.  a — 4 sin3,  a 
3r2  sin.  a — 4 sin2  . a 


or  sin.  3a  = 


R2 


Hence  then,  if  consistently  with  this  precept,  r be  placed 

for  a denominator  of  the  second  member  of  each  equation  v 

(art.  12),  and  if  a be  supposed  equal  to  b,  we  shall  have 

z . \ sin.  a.  cos.  a + sin.  a cos.  a 

sin.  (a  -f-  a)  = : • 


That  is,  sin.  2a  == 


2 sin.  A . cos.  a 
R 


And,  in  like  manner,  by  supposing  b to  become  successively 
equal  to  2a,  3a,  4a,  &c.  there  will  arise 

sin.  A . cos.  2a  -f-cos.  a . sin.  2a.' 


Sin.  3a  = ■ 


sin.  4a 


sin.  a . cos.  Sa+cos.  a . sin.  3a. 


(VIII.) 


sin.  5a  - 


sin.  A . cos.  4a  + cos.  a . sin.  4a' 


And,  by  similar  processes,  the  second  of  the  equations 
just  referred  to,  namely,  that  for  cos.  (a+b),  will  give  suc- 
cessively, 


cos.  2a  — 


cos.  3a  = 
cos.  4a  = 
cos.  5a  = 


COS2  . A — sin2  . A 

sin.  2a 

R 

cos.  a . cos.  2a  — sin.  a . 

R 

cos.  a.  cos.  3a  — sin.  a • 

sin.  3 a 

R 

cos.  a.  cos.  4a— sin.  a . 

sin.  4 a 

(IX.) 


R J 

19.  If,  in  the  expressions  for  the  successive  multiples  of 
the  sines,  the  values  of  the  several  cosines  in  terms  of  the 
sines  were  substituted  for  them  ; and  a like  process  were 
adopted  with  regard  to  the  multiples  of  the  cosines,  other 
expressions  would  be  obtained,  in  which  the  multiple  sines 
would  be  expressed  in  terms  of  the  radius  and  sine,  and  the 
multiple  cosines  in  terms  of  the  radius  and  cosines. 

As  sin.  a = s q 

sin.  2a 


2s^/r3  — s3 
sin.  3a  = 3s — 4s3 


sin.  4a  =(4s  — 8s3)v/r2 — s2 
sin.  5a  = 5s — 20s3  + 16s5 
sin.  6a  =(6s — 32s3  + 32ss).v/r3  — s2 
kc.  kc. 


(X.) 


cos. 


analytical  plane  trigonometry. 


11 


Cos.  A = C 

cos.  2a  = 2c2  — 1 

cos.  3a  = 4c3  — 3c 

cos.  4a  = 8c4  — 8c2  +1  {•  (XI.) 

cos.  5a  = 16c5  — 20c3  -f-  5c 
cos.  6a  = 32c6  — 48c4  -f-  18c2  — 1 
&.c.  &c*. 

Other  very  convenient  expressions  for  multiple  arcs  may 
be  obtained  thus  : 

Add  together  the  expanded  expressions  for  sin.  (b  -f-  a), 
sin.  (b  — a),  that  is, 

add  - - sin.  (b  + a)  = sin.  b . cos.  a + cos.  b . sin.  a, 

to  - - sin.  (b  — a)  = sin.  b . cos.  a — cos.  b.  sin.  a; 

there  results  sin.  (e  + a) -f- sin.  (b  — a)  = 2 cos.  a.  sin.  b: 

whence,  - sin.  (b  -j-  a)  = 2 cos.  a . sin.  b — sin(B  — a). 

Thus  again,  by  adding  together  the  expressions  for  cos.  (b  + A) 
and  cos.  (b— a),  we  have 

cos.  (e  + a)  + cos.  (b  — a)  = 2 cos.  a . cos.  b ; 
whence,  cos.  (b  + a)  = 2 cos.  a . cos.  b — cos.  (b  — a). 
Substituting  in  these  expressions  for  the  sine  and  cosine  of 
b + a,  the  successive  values  a,  2a,  3a,  &c.  instead  of  b ; the 
following  series  will  be  produced. 


sin.  2a  = 2 cos.  a 
sin.  3a  = 2 cos.  a 
sin.  4a  = 2 cos.  a 
sin.  ua  — - 2 cos.  a 
cos.  2a  = 2 cos.  a 
cos.  3a  = 2 cos.  a 
cos.  4a  — 2 cos.  a 


(x.) 


( '**■ ) 


sin.  a. 

sin.  2a  — sin.  a. 
sin.  3a  — sin.  2a. 
sin.  ( re  — 1)  a— sin. (re— 2) a. 
cos.  a — cos.  0 (=1). 
cos.  2a  — cos.  A. 
cos.  3a  — cos.  2a. 
cos.  ua  — 2 cos.  a . cos.  (re  — 1)a  — cos.  (re  — 2) a.  J 
Several  other  expressions  for  the  sines  and  cosines  of  mul- 
tiple arcs,  might  readily  be  found:  but  the  above  are  the 
most  useful  and  commodious. 

..  _ 2 sin  a . cos  a .,  , 

20.  from  the  equation  sin.  2a  = — , it  will  be 

R 

easy,  when  the  sine  of  an  arc  is  known,  to  find  that  of  its 
half.  For,  substituting  for  cos.  a its  value  (r2  — sin2  a), 
,,  „ . 0 2 sin  a (R2  — sin2  a)  „ 

there  will  arise  sm.  2a  = . 1 his  squared 

gives  r2  sin2  2a  = 4r2  sin2  a — 4 sin4  a. 

Here  taking  sin  a for  the  unknown  quantity,  we  have  a quad- 


* Here  we  have  omitted  the  powers  of  R that  were  necessary  to 
render  all  the  terms  homologous,  merely  that  the  expressions  might 
be  brought  in  upon  the  page ; but  they  may  easily  be  supplied,  when 
needed,  by  the  rule  in  art.  18. 


ratic 


It  ANALYTICAL  PLANE  TRIGONOMETRY. 


ratic  equation,  which  solved  after  the  usual  roomer,  gives 

sin  a = ± y/  iR2  ±.  |R  ^ R2  — »in2  2a 
If  we  make  2a  = a',  then  will  a = \a  and  consequently, 
the  last  equation  becomes 


sin 


± N / Ir2  ± ^r  y/  r2  — sin2 


or  sin  \a  = ± a v/2r2  ± 2r.  cos  a'  : 


1 


(XU.) 


by  putting  cos  a'  for  its  value  r2  — sin2  a'  multiplying  the 
quantities  under  the  radical  by  4,  and  dividing  the  whole  se- 
cond number  by  2.  Both  these  expressions  for  the  sine  of 
half  an  arc  or  angle  will  be  of  use  to  us  as  we  proceed. 

21.  If  the  values  of  sin  (a  + b)  and  sine  (a  — b),  given  by 
fequa.  v,  be  added  together,  there  will  result 

. . . . , . . . 2 sin  a . cos  b , 

sin  (a  + b)  + sin  (a  — b)  = — ; whence, 

\ \ / r 

sin  a . cos  b = -Jr  sin  (a  + e)  + i r sin  (a  — b)  . (XIII.) 
Also,  taking  sin  (a  — b)  from  sin  (a  + b)  gives 


sin  (a  + b)  — sin  (a  — b)  = 


2 sin  b . c s a 


whence, 


sin  b . cos  a =4r  sin  (a+b)  — Ar  . sin  (a — b)  . . (XIV.) 
When  a — b both  equa.  xm  and  xiv,  become 
cos  a . sin  a = Ar  sin  2a  . . (XV.) 

22.  In  like  manner,  by  adding  together  the  primitive  ex- 
pressions for  cos  (a  + b),  cos  (a  — b),  there  will  arise 

, , n . , s 2 c«.s  a . cos  b , 

cos  (a  + b)  + cos  (a  — b)  = — ; whence, 

COS  A . COS  B ==  Ar  . cos  (a  + b)  + Ar  . C0S  (a b)  (XVI.) 

And  here,  when  a = b,  recollecting  that  when  the  arc  is 
nothing  the  cosine  is  equal  to  radius,  we  shall  have 
cos2  a — |r  . cos  2a  -f-  Ar2  . . . (XVII.) 

23.  Deducting  cos  (a  + b)  from  cos  . (a  — b),  there  will 
remain 

, . /iv  2 sin  a . sin  b , 

cos  (a  — b)  — cos.  (a  + b)  = ^ ; whence, 

sin  a . sin  b=  ^r  . cos  (a  — b)  — ^r  . cos  (a+b)  (XVIII.) 
When  a = b,  this  formula  becomes 

sin2  A=  Ar2  __  Ar  . cos  2a  . . . (XIX.) 

24.  Multiplying  together  the  expressions  for  sin  (a  + b) 
and  sin  (a  — b),  equa.  v,  and  reducing,  there  results 

■ sin  (a  + b)  . sin  (a  — b)  ==-  sin2  a — sin2  b. 

And,  in  like  manner,  multiplying  together  the  values  of  cos 
(a  + b)  and  cos  (a  — b),  there  is  produced 

cos  (a  + b)  . cos  (a  — b)  = cos2  a — cos2  b. 

Here,  since  sin2  a — sin2  b,  is  equal  to  (sin  a + sin  b)  X 
fsin  a — sin  b),  that  is,  to  the  rectangle  of  the  sum  and  dif- 
ference 


ANALYTICAL  PLANE  TRIGONOMETRY. 


13 


ference  of  the  sines  ; it  follows,  that  the  first  of  these  equa- 
tions converted  into  an  analogy,  becomes 

siu  (a— b)  : sin  a - sin  b : : sin  a 4-  sin  b : sin  (a  + b)  (XX.) 
That  is  to  say,  the  sine  of  the  difference  of  any  two  arcs  or 
angles,  is  to  the  difference  of  their  sines,  as  the  sum  of  ihose  sines 
is  to  the  sine  of  their  sum. 

If  a and  b be  to  each  other  as  n -f-  1 to  n,  then  the  preced- 
ing proportion  will  be  converted  into  sin  a : sin  (re  + 1)  a — 
sin  TiA  : : sin  (re  + 1)  a + sin  nx  : sin  (2re  + 1)  a . . . . (XXI.) 

These  two  proportions  are  highly  useful  in  computing  a ta- 
ble of  s'ines  ; as  will  be  shown  in  the  practical  examples  at  the 
end  of  this  chapter. 

25.  Let  us  suppose  a + b = a',  and  a — b = b'  ; then  the 
half  sum  and  the  half  difference  of  these  equations  will  give 
respectively  a = a(a'-{-b'),  and  b = a(a'_b').  Putting  these 
values  of  a and  b,  in  the  expressions  of  sin  a . cos  b,  sin  b . cos  a, 
cos  a . cos  b,  sin  a . sin  b,  obtained  in  arts.  2],  22,  23,  there 
would  arise  the  following  formula;  : 

sin  A (a'+b')  . cos  a(a'  — b')  = -^R(sin  A'+sin  b'), 
sin  a (a'  — b')  . cos  a(a'-)-b')  = iji(sin  a'  —sin  b), 
cos  a (a'+b')  . cos  a(a'  - b ) — ar(cos  a'  + cos  b ), 
sin  A (a'+b')  sin  a(a'  — b')=  4-r(cos  b'~  cos  a ). 
Dividing  the  second  of  these  formulae  by  the  first,  there  will 
» be  had 


sin|(A  — 3 ) cos^A't-B  ) sinjfA'-  b')  cos|(a'  + b') sinA-  sinn' 

3in^(A  + b ) ’ cosi(A'— b')  cos|(A  -B  )"sin^(A  +B)  siha' -f  suib' 
tan  . cos  R .,  „ ,, 

— , and—  = - — , it  follows  that  the  two 
r sin  tan 

factors  of  the  first  member  of  this  equation,  are 
tanif  a'—  b)  , R , 

and  - — : - — -.respectively  ; so  that  the  equation 


But  since  — 

cos 


tan^(A+B  ) 

manifestly  becomes  Un 


sin  a —sin  b 


..  (XXII.) 


tan  ,J(a  -(-  B ) sin  a'  -f-  sm  b' 

This  equation  is  readily  converted  into  a very  useful  pro- 
portion, viz.'  The  sum  of  the  sines  of  two  arcs  or  angles,  is  to 
their  difference,  as  the  tangent  of  half  the  sum  of  those  arcs  or 
angles,  is  to  the  tangent  of  half  their  difference. 

26.  Operating  with  the  third  and  fourth  formulae  of  the 
preceding  article,  as  we  have  already  done  with  the  first  and 
second,  we  shall  obtain 

tan  ^ (a'  4 b')  . tan  ^(a'  - b ) cos  b'  — cos  a' 


R2  cos  a1  + cos  b' 

In  like  manner,  we  have  by  division, 

sirA'f hi'b'  sin^(A'  + B')  ‘ sinA' + sinB' 

7-t 7= — i7+T-%=tani(A +B  ); -=coO.(a  -b  ); 

COS  A + COS  B COsi(A  -f-B  ) ' COSB  — COSA  2V 

sinA  — sinB'  , , ,N  sin  a'- - in  b' 

COSA-f-COSB  1 ' ' COSB  — COSA  “ V ' 


cos 


14 


ANALYTICAL  PLANE  TRIGONOMETRY. 


COS  a'  + COS  b'  cot  $ (a’  + b') 

cos  b'  — cos  a'  tan  ^ (a'  — b')’ 

Making  b = 0,  in  one  or  other  of  these  expressions,  there 
results, 


l-j-  COS  A 

sin  a' 

1 — cos  a' 


— , — tan  4 a'  = 


1 


cot  ^a' 
, 1 
= cot  4a  = — 


tan  ^a'  * 

1 + cos  a'  cot  A a'  ..  , , 1 

; = — , = COt2  iA  = 

. tan  ^ A 2 


( xxii.j 


1 — cos  a'  tan  ^ a'  2“  tan2  ^a’’ 

These  theorems  will  find  their  application  in  some  of  the 
investigations  of  spherical  trigonometry. 

27.  Once  more,  dividing  the  expression  for  sin  (a  ± b) 
by  that  for  cos  (a  ± b),  there  results 

sin  (a  ■+:  b)  sin  a . cos  e + sin  b . cos  a 

cos  (a  ± b)  cos  a . cos  b sin  a . sin  b 

then  dividing  both  numerator  and  denominator  of  the  second 


tan 


fraction  by  cos  a . cos  b,  and  recollecting  that  — ^ = 
shall  thus  obtain 

tan  (a  b) 


we 


or,  lastly,  tan  (a  ± b) 


_ r (tan  a ± tan  b). 

r2  +tan  A . tan  b) 
r2  (tan  a ±;  tan  b) 


tan  a . tan  b 


(XXIII.) 


Also,  since  cot  = — , we  shall  have 
tan 

cot  (a  ± b)  = 

' ' 1 


1*  R2  -t-  tan  a - tan  b 

tan  (a  ± b)  tan  a ± tan  b 

w'hich,  after  a little  reduction,  becomes 

cot  (A  ± BJ  = ....  (XX1\  .) 

V ' COt  B -+-  COt  A V ' 


28.  We  might  now  proceed  to  deduce  expressions  for  the 
tangents,  cotangents,  secants,  &c.  of  multiple  arcs,  as  well  as 
some  of  the  usual  formulas  of  verification  in  the  construction  of 
tables,  such  as 

sin(54°+A)  + sin(54°  — A)~sin(18Q  + A)~sin(18°—  a)  = sin(90°  — A)  ; 
sin  a +sin(36°  — A)  + sin(72°+  A)=sin(36°  + a)  -hsin(72°  — a). 

&c.  &c. 

But,  as  these  enquiries  would  extend  this  chapter  to  too 
great  a length,  we  shall  pass  them  by  ; and  merely  investigate 
a few  properties  where  more  than  two  arcs  or  angles  are  con- 
cerned, and  which  may  be  of  use  in  some  subsequent  part  of 
this  volume. 


29.  Let 


analytical  plane  trigonometry. 


15 


29.  Let  a,  b,  c,  be  in  any  three  arcs  or  angles,  and  suppose 
radius  to  be  unity  ; then 

sin  a . sin  c -f-  sin  b . sin  (A  + B + c). 
sin  (a  -f-  b) 

For,  by  equa.  v,  sin  (a+b+c)  = sin  a . cos  (b+c)  -j-  cos  a . 
sin  (b  + c),  which,  (putting  cos  b . cos  c — sin  b . sin  c for 
cos  (b  + c)),  is  = sin  a . cos  b . cos  c — sin  a . sin  b . sin  c + 
cos  a . sin  (b  + c)  ; and,  multiplying  by  sin  b,  and  adding 
sin  a . sin  c,  there  results  sin  a . sin  c + sin  b . sin  (a+b+c) 
= sin  a . cos  b . cos  c . sin  b -j-  sin  a . sin  c . cos2  b + cos  a . 
sin  b . sin  (b+c)  = sin  a . cos  b . (sin  b . cos  c-J-ecs  b . sin  c) 
-f-  cos  a . sin  b . sin  (b  -f-  c)  = (sin  a . cos  b -f*  cos  a . sin  b)  X 
sin  (b  + c)  = sin  (a  -f-  b)  . sin  (b  + c).  Consequently,  by 
dividing  by  sin  (a  -f-  b),  we  obtain  the  expression  above 
given. 


sin  (b  -pc)  — 


In  a similar  manner  it  may  be  shown,  that 

sin  A . sin  c — sin  B • sin  (A  — b + c) 
sin  (a  — b) 


sin  (b  — c)  = 


30.  If  a,  b,  c,  d,  represent  four  arcs  or  angles,  then  writ- 
ing c+d  for  c in  the  preceding  investigation,  there  will 
result, 

sin  A . sin(c  -f-  D + sin  E . sin(A-j-B  + c +d) 
sin  (a  -f-  b) 

A like  process  for  five  arcs  or  angles  will  give 


sin  (b-J-c+d)  = 


sin(B+c-{-D-l-E): 


_sinA.sin(c  4-  D-fEld- sinB.«in(A -J-  B+C-f  p + e). 
sin  (a  + b) 

And  for  any  number,  a,  b,  c,  &lc.  to  l, 

....  . sinA.sinfc -l-D  + ~L)  + sinB.sin(A-f-  B-f-  C-j — l) 

sm(B+c+....L)= lilTc r+i) 


31.  Taking  again  the  three  a,  b,  c,  we  have 
sin  (b  — c)  = sin  b . cos  c— sin  c . cos  b, 

sin  (c— a)  = sin  c . cos  a— sin  a . cos  c, 

sin  (a— b)  = sin  a . cos  b — sin  b . cos  a. 

Multiplying  the  first  of  these  equations  by  sin  a,  the  second 
by  sin  b,  the  third  by  sin  c ; then  adding  together  the  equa- 
tions thus  transformed,  and  reducing  ; there  will  result, 
sin  a . sin  (b  — c)+sin  b . sin  (c  — a)  + sin  c . sin  (a—  b)  = 0, 

cos  a . sin  (b— c)+cos  b . sin  (c— a)  -j-  cos  c . sin  (a  — b)  =0, 


These  two  equations  obtaining  for  any  three  angles  what- 
ever, apply  evidently  to  the  three  angles  of  any  triangle. 

32.  Let  the  series  of  arcs  or  angles  a,  b,  c,  d . . . . t,  be 
contemplated,  then  we  have  (art.  24), 

sin 


16 


ANALYTICAL  PLANE  TRIGONOMETRY- 


sin  (a  + b)  . sin  (a  — b)  = sin2  a — sin2  b, 

sin  (b  + c)  . sin  (b  — c)  = sin2  b — sin2  c, 

sin  (c  -f*  d)  . sin  (c—  d)  = sin2  c — sin2  d, 

&c.  &c.  &c. 

sin  (l  + a . sin  (l  — a)  = sin2  l— sin2  a. 

If  all  these  equations  be  added  together,  the  second  member 
of  the  equation  will  vanish,  and  of  consequence  we  3hall  have 
sin  (a+b)  . sin  (a  — b)  + sin  (b+c)  . sin  (b  — c)  + &c.  . . 

+ sin  (l+a)  + sin  (l—  a)  = 0. 

Proceeding  in  a similar  manner  with  sin  (a  — b),  cos  (a  + b), 
sin  (b  — c),  cos  (b+c),  &c.  there  will  at  length  be  obtained 
cos  (a+b)  . sin  (a-'b)  + cos  (b+c)  . sin  (b — c)  + &c . . . 
+ cos  (l+a)  . sin  (l—  a)  = 0. 

33.  If  the  arcs,  a,  b,  c,  &c l form  an  arithmetical 

progression,  of  which  the  first  term  is  0,  the  common  differ- 
ence d\  and  the  last  term  l any  number  n of  circumferences  ; 
then  will  b — a = d',  c — b --  d',  &c.  a + b = d',  b + c = 3d', 
&c.  : and  dividing  the  whole  by  sin  d',  the  preceding  equa- 
tions will  become 

sin  d'  + sin  3d'  + sin  5d'  + &c.=  0,  } v 

cos  d'  + cos  3d'  + cos  5d'  + &c.  = 0.  $ ^ 

If  e'  were  equal  2d',  these  equations  would  become 

sin  d’+ sin  (n'  + E')+sin  (d-^e')  + sin  (d +3e')+ &c.  = 0, 

cos  d'+  cos  (d'+e')  + cos  (d  +2e')  + cos(d'+3e')+  kc.  = 0. 


34.  The  last  equation,  however,  only  shows  the  sums  of 
sines  and  cosines  of  arcs  or  angles  in  arithmetical  progres- 
sion, when  the  common  difference  is  tb  the  first  term  in  the 
ration  of  2 to  1.  To  investigate  a general  expression  for  an 
infinite  series  of  this  kind,  let 

s + sin  a + sin  (a  + b)  + sin  (a  + 2b)  sin  (a  +3b)  + kc. 
Then,  since  this  series  is  a recurring  series,  whose  scale  of 
relation  is  2 cos  b — 1,  it  will  arise  from  the  developement  of 
a fraction  whose  denominator  is  1 — 2z  . cos  b + z2 , making 
2=1.  \ 


Now  this  fraction  will  be 


sin A+2[sin(A+B)  — 2 sin  A . cos  b] 
1 — 2 2 • cos  b +s2 


Therefore,  when  2=1,  we  have 


sin  A + sin  (a  + Bl  — 2 sin  a . cos  b , ,,  . , _ . 

s = ^ — ; and  this, because  2sin  a, 

2—2  cos  b 

cos  b = sin  (a  + b)  + sin  (a  — b)  (art.  21),  is  equal  to 

sinA— sin(A  — b)  , • , _ , , . , 

— 2 -.  But,  since  sin  a—  sin  b = 2 cos  +a  + b ). 

2(1  — cos  b)  J 


sin 


ANALYTICAL  PLANE  TRIGONOMETRY. 


17 


(XXVLi.) 


sin  a(a'— b'),  by  art.  25,  it  follows,  that  sin  a — sin  (a  — b)  = 
2 cos  (a  — ab)  sin  ab  ; besides  which  we  have  1 — cos  b = 
2 sin2  iB.  Consequently  the  preceding  expression  becomes 
s — sin  a -{-  sin  (a  4 b)  4"  sin  (a  4 2b)  4 sin  (A  4"  3b)  4 &c> 

ad  infinitum  = (XXVI.) 

35.  To  find  the  sum  of  n 4 1 terms  of  this  series,  we  have 
simply  to  consider  that  the  sum  of  the  terms  past  the  (w+  1)  th, 
that  is,  the  sum  of  sin  [a  4 (n  4 1)  b]  4 sin  [a  + (n  4 2)b]4 
sin  [a  -f-  (n  4 3)  b]  -f-  &c.  ad  infinitum,  is,  by  the  preceding 

theorem,  =r-s.IA.-  ■ . Deducting  this,  therefore,  from 

2*iMb  . 

the  former  expression,  there  will  remain,  sin  a 4 sin  (a  4 b) 
+ sin  (a  -f  2b)  4 sin  (a  4 3b)  4 ....  sin  (a  4 «b)  = 
cos(a  — ^b )™ cos[ a-P1  n~h  Y B 1 sin(A4  inBVin^(7i+ ' ,b 
2 sin  ^b  sin 

By  like  means  it  will  be  found,  that  the  sums  of  the 
©osines  of  arcs  or  angles  in  arithmetical  progression  will  be 
cos  a -f*  cos  (a  4 b)  4 cos  (a  4-  2b)  4-  cos  a 4"  3b)  4 &c. 

ad  infinitum  = — (XXVIII.) 

Also, 

COS  A 4-  COS  (a  4 b)  4 cos  (a  + 2b)  4-  cos  (a  4-  3b)  4-  • . - . 

....  (COS  A + »B)  .«»<»+>.  »>•*■«<■♦».  . . (XXIX.) 

36.  With  regard  to  the  tangents  of  more  than  two  arcs, 

the  following  property  (the  only  one  we  shall  here  deduce.)  is 
a very  curious  one,  which  has  not  yet  been  inserted  in  works 
of  Trigonometry,  though  it  has  been  long  known  to  mathe- 
maticians. Let  the  three  arcs  a,  b,  c,  together  make  up 
the  whole  circumference,  O : -then  since  tan  (a  4-  b)  = 
H2  (tan  A 4 tan  b).,  , , „ . , , , , 

| ■ '(by  equa.  xxm ),  we  have  r2  X (tan  a 4 tan  b 4 

r2  _ tan  a • tan  b w n J K * 

tan  c)  = r2  X [tan  a 4 tan  b — tan  (a  4*  b)]  = r2  X (tan  a 4 

. r2  (tan  a -f-  tan  B).  , ...  ..  ,.  , 

tan  b — — r -)=,  by  actual  multiplication  and  re- 

R “ ~ tan  pl  • tan  b 

duction,  to  tan  a . tan  b . tan  c,  since  tan  c = tan  [ Q — 

R2  (tan  a + tan  b ) , , 

— -,  by  what  has 


4a  4 b)1  = — tan  (a  — b) 

^ n K J R2  — tan  A • tan  b' 

preceded  in  this  article.  The  result  therefore  is,  that  the 
sum  of  the  tangents  of  any  three  arcs  -which  together  consti- 
tute a circle,  multiplied  by  the  square  of  the  radius,  is  equal 
to  the  product  of  those  tangents.  ....  (XXX.) 

Since  both  arcs  in  the  second  and  fourth  quadrants  have 
their  tangents  considered  negative,  the  above  property  will 
apply  to  arcs  any  way  trisecting  a semicircle  ; and  it  will  there- 
, Von.  II.  4 fore 


18  ANALYTICAL  PLANE  TRIGONOMETRY. 


fore  apply  to  the  angles  of  a plane  triangle,  which  are 
together,  measured  by  arcs  constituting  a semicircle.  So 
that  if  radius  be  considered  as  unity,  we  shall  find  that,  the 
sum  of  tangents  of  the  three  angles  of  any  plane  triangle , is 
equal  to  the  continued  product  of  those  tangents.  (XXXI  ) 

37.  Having  thus  given  the  chief  properties  of  the  sines, 
tangents,  &c.  of  arcs,  their  sines,  products,  and  powers,  we 
shall  merely  subjoin  investigations  of  theorems  for  the  2d  and 
3d  cases  in  the  solutions  of  plane  triangles.  Thus,  with  re- 
spect to  the  second  case,  where  two  sides  and  their  included 
angle  are  given  : 


By  equa.  iv,  a : b : : sin  a : sin  b. 

and  "division  | a + b : a — b : : sin  a -f  sin  b : sin  a — sin  e. 

But,  eq.  xxii,  tan  1 (a  -f-  b)  : tan  i (a  — b)  : : sin  a -f-  sin  B 
sin  a—  sin  b ; whence,  ex  equal  a -f-  b : a — 6 : : tan  4(a  + b)  : 
tan  £(a  — b) (XXXII.) 

Agreeing  with  the  result  of  the  geometrical  investigation, 
at  pa.  386,  vol.  i. 


38.  If,  instead  of  having  the  two  sides  a,  b,  given,  we  know 
their  logarithms,  as  frequently  happens  in  geodesic  opera- 
tions, tan  a (a — b)  may  be  readily  determined  without  first 
finding  the  number  corresponding  to  the  logs  of  a and  b . 
For  if  a and  b were  considered  as  the  sides  of  a right-angled 
triangle,  in  which  <p  denotes  the  angle,  opposite  the  side  a, 

then  would  tan  <p  = Now,  since  a is  supposed  greater 
b 


than  b , this  angle  will  be  greater  than  half  a right  angle,  or 
it  will  be  measured  by  an  arc  greater  than  | of  the  circumfer- 

i mi  i . _ , tin  p —t^n  i Q 

ence,  or  than  AQ-  Then,  because  tan  (<p  — 40)  =r-; — — ~~ 

- 8 ' 1-funptanJ-O 

and  because  tan  a Q = R = 1,  we  have 

io) =(;->)  -(>+;-= £„■ 

And,  from  the  preceding  article, 

t i—b  tan  s(a — b)  tan  £(A  — b) 

— T7= — 77  {— ^ -•  consequently, 

a+6  tail  $(a  + B)  coi 

tan -’-(a — b)  = cot  ic  . tan  (<p  — a Q)  . . . (XXXIII) 
From  this  equation  we  have  the  following  practical  rule. 
Subtract  the  less  from  the  greater  of  the  given  logs,  the  re- 
mainder will  be  the  log  tan  of  an  angle  : from  this  angle 
take  45  degrees,  and  to  the  log  tan  of  the  remainder  add  the 
log  catan  of  half  the  given  angle  ; the  sum  will  be  the  log 
tan  of  half  the  difference  of  the  other  two  angles  of  the  plane 
triangle. 

39.  The 


analytical  plane  trigonometry. 


19 


39.  The  remaining  case  is  that  in  which  the  three  sides  of 
the  triangle  are  known,  and  for  which  indeed  we  have  al- 
ready dbtained  expressions  for  the  angles  in  arts.  6 and  8. 
But,  as  neither  of  these  is  best  suited  for  logarithmic  compur 
tation,  (however  well  fitted  they  are  for  instruments  of  in- 
vestigation), another  may  be  deduced  thus  : in  the  equation 

for  cos  a,  (given  equation  ii),  viz.  cos  a = b ' — , if  we 


substitute,  instead  of  cos  a,  its  value,  1 — 2 sin3  |a,  change 
the  signs  ot  all  the  terms,  transpose  the  1,  and  divide  by  2, 

. n , • „ , — c4  4-  26c  a2-(b-c)2 

we  shall  have  sin2  4a  = — = 

46c  46c 

Here,  the  numerator  of  the  second  member  being  the  pro- 
duct of  the  two  factors  (a  -f-  b — c)  and  (a  — b -f-  c),  the  equa- 
tion will  become  sin2  4a  = j(.fl  — 64~c),  sjnce 

46c 

i(a-f-6 — c)  = £(a+&-t-c)  — c,andi(a  — 6+c)=i-(a+6-{-c) — b ; 
if  we  put  s = a -f-  b -j-  c,  and  extract  the  square  root,  there 
will  result, 


In  like  ) 
manner  ) 


sin  1a  = ^ 
sin  4.3  = ^/ 
sin  4c  = 


(js— 6)  ■ (js — c) 
be 

(,js -a)  ■ (js  — c) 
ac 

(\s-a)  ■ (£->-6) 
ab 


(XXXIV.) 


These  expressions,  besides  their  convenience  for  logarith- 
mic computation,  have  the  further  advantage  of  being  perfect- 
ly free  from  ambiguity,  because  the  half  of  any  angle  of  a 
plane  triangle  will  always  be  less  than  a.  right  angle. 


40.  The  student  will  find  it  advantageous  to  collect  into 
one  place  all  those  formulas  which  relate  to  the  computation 
of  sines,  tangents,  &c.*  ; and,  in  another  place,  those  which 
are  of  use  in  the  solutions  of  plane  triangles  : the  former  of 
these  are  equations  v,  vm,  ix,  x,  xi,  x,  xi,  xn,  xm,  xiv, 
xv,  xvi,  xvii,  xvm,  xix,  xx,  xxii,  xxii,  xxm,  xxiv, 
xxvii  ; the  latter  are  equa.  ii,  hi,  iv,  vn,  xxxn,  xxxm, 
xxxiv. 

To  exemplify  the  use  of  some  of  these  formulae,  the  follow- 
ing exercises  are  subjoined. 


* What  is  here  given  being  only  a brief  sketch  of  an  inexhausti- 
ble subject  ; the  reader  who  wishes  to  pursue  it  further  is  referred 
to  the  copious  Introduction  to  our  Mathematical  Tables,  and  the 
comprehensive  treatises  on  Trigonometry,  bv  Emerson  and  many 
other  modern  writers  on  the  same  subject,  where  he  will  find  his 
curiosity  richly  gratified. 


EXERCISES. 


20  ANALYTICAL  PLANE  TRIGONOMETRY 


EXERCISES. 

Ex.  1.  Find  the  sines  and  tangents  of  15°,  30°,  45°,  60°. 
and  75°  : and  show  how  from  thence  to  find  the  sines  and 
tangents  of  several  of  their  submultiples. 

First,  with  regard  to  the  arc  of  45°,  the  sine  and  cosine  are 
manifestly  equal  ; or  they  form  the  perpendicular  and  base 
of  a right  angled  triangle  whose  hypothenuse  is  equal  to  the 
assumed  radius.  Thus,  if  radius  be  r,  the  sine  and  cosine  of 
45°,  will  each  be  = ^/  ar2  = r^/a  = ^r^/2.  If  r be  equal 
to  1,  as  is  the  case  with  the  tables  in  use,  then 
sin  45°  = cos  45°  - 4^/2  = -7071068. 

tan  45°  = — = 1 = ^-S  = cotangent  45°. 

cos  sin  ° 

Secondly,  for  the  sines  of  60°  and  of  30°  : since  each  angle 
in  an  equilateral  triangle  contains  60°,  if  a perpendicular  be 
demitted  from  any  one  angle  of  such  a triangle  on  the  oppo- 
site side,  considered  as  a base,  that  perpendicular  will  be  the 
sine  of  6u°,  and  the  halfbase  the  sine  of  30°,  the  side  of  the 
triangle  being  the  assumed  radius.  Thus,  if  it  be  r,  we  shall 
have  iR  for  the  sine  of  30°,  and  ^r2  — ar2  = ^R^/Sjfor  the 
sine  of  60°.  When  r = 4,  these  become 

sin  30°  = -5 sin  60°  = cos  30°  = -8660254. 

Hence,  tan  30°  ==  = —L.  =^3  = -577S503, 


tan  60° 


W3 
i\/3 


v/2 

= V*  =• 


1-7320508. 


Consequently,  tan  604  = 3 tan  30°. 

Thirdly,  for  the  sines  of  15°  and  75°,  the  former  arc  is  the 
half  of  30°,  and  the  lather  is  the  compliment  of  that  half  arc. 
Hence,  substituting  1 for  r and  \^/3,  for  cos  a,  in  the  ex- 
pression sin  |a  = ± a y/  2r2  ± 2r  cos  a . . . (equa.  xn), 
it  becomes  sin  15°  — 2 — 3 = -258S190. 

Hence,  sin  75°  = cos  15°  = y/\ — a (2 — <y3)  — ±^/2-{-^/3  — 


^y'v/-  = -9659258. 

4 

Consequently,  tan  1513  = — = 


•2588190 

•9659258 


= -2679492. 


And,  tan  75°  = 


•9659258 


= -3-7320508. 


'2588190 

Now,  from  the  sine  of  30°',  those  of  6°,  2°.  and  1°,  may 
easily  be  found.  For,  if  5a  = 30°,  we  shall  have,  from 
equation  x,  sin  5a  = 5 sin  a — 20  sin3  a+ 16  sin5  a : or,, if 
sin  a — x,  this  will  become  16x5  — 20x3  + 5x  = -5  This 
equation  solved  by  any  of  the  approximating  rules  for  such 
equations,  will  give  x = -1045285,  which  is  the  sine  of  6 °. 

Next 


analytical  plane  trigonometry. 


21 


Next,  to  find  the  sine  of  2°,  we  have  again,  from  equation 
x,  sin  3a  = 3 sin  a—  4 sin3  a : that  is,  if  x be  put  for  sin  2°, 
3a;  — 4x3  = TQ45285.  This  cubic  solved,  gives  x — -0348995 
= sin  2°. 

Then,  if  s = sin  1°,  we  shall,  from  the  second  of  the  equa- 
tions marked  x,  have  2s  1— s2  .=  -0348995  ; whence  s is 

found  = -0174524  = sin  1°. 

Had  the  expression  for  the  sines  of  bisected  arcs  been  ap- 
plied successively  from  sin  15°,  to  sin  7°30',  sin  3°45',  sin 
l°52i'  sin  56i',  &c.  a different  series  of  values  might  have 
been  obtained  : or,  if  we  had  proceeded  from  the  quinqui- 
section  of  45°,  to  the  trisection  of  9°,  the  bisection  of  3°,  and 
so  on,  a different  series  still  would  have  been  found.  But 
what  has  been  done  above,  is  sufficient  to  illustrate  this  method. 
The  next  example  will  exhibit  a very  simple  and  compendious 
way  of  ascending  from  the  sines  of  smaller  to  those  of  larger 
arcs. 

Ex.  2.  Given  the  sine  of  1°,  to  find  the  sine  of  2°,  and 
then  the  sines  of  3°,  4°,  5°,  6°,  7°,  8°,  9°,  and  10°,  each  by  a 
single  proportion. 

Here,  taking  first  the  expression  for  the  sine  of  a double 


arc,  equa.  x,  we  have  sin  2°  = 2sin  l°y/\  —sin2  l°  = -034895. 
Then  it  follows  from  the  rule  in  equa.  xx,  that 


sin  1° 

sin  2° 

— sin  1° 

: sin  2°  -f-  sin  1° 

sin 

3°  = -0523360 

sin  2° 

sin  3Q 

— sin  1 J 

: sin  3°  -|-  sin  1° 

sin 

4°  = -0697565 

sin  3° 

sin  4° 

— sin  1° 

: sin  4°  -j-  sin  1° 

sin 

5°  = 0871557 

sin  4° 

sin  b° 

— sin  1° 

: sin  5“  — {—  sin  1° 

sin 

6°  = • 1Q45285 

sin  5° 

sin  6° 

—sin  1G 

; sin  6°  + sin  1° 

sin 

7°  = -1218693 

sin  6° 

sin  7° 

— sin  1° 

: sin  7°  -j-  sin  1° 

sin 

8°  = -1391731 

sin  7° 

sin  8° 

—sin  1° 

: sin  8°  -|-  sin  1° 

sin 

9°  = -1564375 

sin  8° 

sin  9° 

— sin  1° 

: sin  9°  -{-sin  1° 

sin!0°  = -1736482 

To  check  and  verify  operations  like  these,  the  proportions 
should  be  changed  at  certain  stages.  Thus, 

sin  1°  : sin  3°  — sin  2°  : : sin  3°  + sin  2°  : sin  5°, 

sin  1°  : sin  4°  — sin  3°  : : sin  4°  -j-  sin  3Q  : sin  7°, 

sin  4°  : sin  7°— sin  3°  : : sin  7°  4*  sin  3°  : sin  10°. 

The  coincidence  of  the  results  of  these  operations  with  the  ana- 

logous results  in  the  preceding,  will  manifestly  establish  the 
correctness  of  both. 

Cor.  By  the  same  method,  knowing  the  sines  of  5°,  10°, 
and  15°,  the  sines  of  20°,  25°,  35°,  55°,  65°,  &c.  may  be 
found,  each  by  a single  proportion.  And  the  sines  of  1°,  9°, 
and  10°,  will  lead  to  those  of  19®,  29°,  39°,  k.c.  So  that  the 
sines  may  be  computed  to  any  arc  : and  the  tangents  and  other 
trigonometrical  lines,  by  means  of  the  expressions  in  art.  4,  &c. 


22 


ANALYTICAL  PLANE  TRIGONOMETRY. 


Ex.  3.  Find  the  sum  of  all  the  natural  sines  to  every  mi- 
nute in  the  quadrant,  radius  = 1 

In  this  problem  the  actual  addition  of  all  the  terms  would 
be  a most  tiresome  labour  : but  the  solution  by  means  of 
equation  xxvii,  is  rendered  very  easy.  Applying  that  theo- 
rem to  the  present  case,  we  have  sin  (a  -f-  \n  b)  = sin  45°, 
sin  \{n  -p  1)  B=sin  45°0'30 ',  and  sin  4 B=sin  30".  Therefore 

sin  45°  x sin  45°  0'  30 ' «j,0  j 

77 = 3438-24b746o  the  same  sum  required. 

sin  30  n 

From  another  method,  the  investigation  of  which  is  omitted 
here,  it  appears  that  the  same  sum  is  equal  to  4 (cot  30 ' + 1). 

Ex.  4.  Explain  the  method  of  fiuding  the  logarithmic, 
sines,  cosines,  tangents,  secants,  &c.  the  natural  sines,  cosines, 
Sic.  being  known. 

The  natural  sines  and  cosines  being  computed  to  the  radiu6 
unity,  are  all  proper  fractions,  or  quantities  less  than  unity, 
so  that  their  logarithms  would  be  negative.  To  avoid  this, 
the  tables  of  logarithmic  sines,  cosines,  Sic.  are  computed  to 
a radius  of  10000000000,  or  1010  : in  which  case  the  loga- 
rithm of  the  radius  is  10  times  the  log  of  10,  that  is,  it  is  10. 

Hence,  if  s represent  any  sine  to  radius  1,  then  1010  Xs  = 
sine  of  the  same  arc  or  angle  to  rad  1010.  And  this,  in  logs 
is,  log.  10l0s  = 10  log.  10  -f-  log.  s = 10  + log.  s. 

The  log  cosines  are  found  by  the  same  process,  since  the 
cosines  are  the  sines  of  the  complements. 

The  logarithmic  expressions  for  the  tangents,  Sic.  are  de- 
duced thus  : • 

sin 

Tan  = rad  — . Theref.  log  tan  ==  log  rad  -{-  log  sin  — log 

cos  = 10  -f-  log  sin  — log  cos. 

2 

Cot  = Therf.  log  cot=2  log  rad  - log  tan=20— log  tan. 

rad2 

Sec  =-^-.  Therf.  log  sec=2  log  rad  — log cos=  20 -log  cos. 
rad 

Cosec  =-7jp  Therf.l.cosec=21ograd— logsin=20  — logsin. 

, chord1  (2  sin  4 arc)2  2 X sin2  A arc 

diam  2 rail  rad 

Therefore,  log  vers  sin  = log  2 -f-  2 log  sin  4 arc  — 10. 

Ex.  5.  Given  the  sum  of  the  natural  tangents  of  the  an 
gles  a and  b of  a plane  triangle  — 3-1601988,  the  sum  of  the 
tangents  of  the  angles  b and  c = 31  8765577,  and  the  conti- 
nued product,  tan  a . tan  b . tan  c = 5-3047057  : to  hnd  the 
angles  a,  b,  and  c. 

It 


ANALYTICAL  PLANE  TRIGONOMETRY. 


It  has  been  demonstrated  in  art.  36,  that  when  radius  is 
unity,  the  product  of  the  natural  tangents  of  the  three  angles 
of  a plane  triangle  is  equal  to  their  continued  product.  Hence 
the  process  is  this  : 

From  tan  a + tanB  -{-  tan  c = 5-3047057 

Take  tan  a tan  b . . . . = 3-1601988 

Remains  tan  c = 2-1445069  = tan  65° 

From  tan  a -{-  tan  b -f-  tan  c = 5-3047057 

Take  tan  b -f-  tan  c . . . . = 3-8765577 

Remains  tan  a = 1 -428 1480  = tan  55°. 

Consequently,  the  three  angles  are  55°,  60°,  and  65°. 

Ex.  6.  There  is  a plane  triangle,  whose  sides  are  three 
consecutive  terms  in  the  natural  series  of  integer  numbers, 
and  whose  largest  angle  is  just  double  the  smallest.  Requir- 
ed the  sides  and  angles  of  that  triangle  ? 

If  a.  b,  c,  be  three  angles  of  a plane  triangle,  a,  b,  c,  the 
sides  respectively  opposite  to  a,  b,  c ; and  s = a -|-  b -{-  c. 
Then  from  equa.  hi  and  xxxiv,  we  have 


Let  the  three  sides  of  the  required  triangle  be  represented 
by  x,  x -f-  1,  and  x -f-  2 ; the  angle  a being  supposed  oppo- 
site to  the  side  x,  and  c opposite  to  the  side  x -j-  2 : then  the 
preceding  expressions  will  become 


Assuming  these  two  expressions  equal  to  each  other,  as  they 
ought  to  be,  by  the  question  ; there  results,  after  a little  re- 
duction. x3—%x2  V1  x 2 = 0,  a cubic  equation,  with  one 
positive  integer  root  x — 4.  Hence  4,  5,  and  6,,are  the  sides 
of  the  triangle. 


The  angles  are,  a = 41°-409603=  41°24'  34"jj  34 


Any  direct  solution  to  this  curious  problem,  except  by  means 
of  the  analytical  formulae  employed  above,  would  be  exceed- 
ingly tedious  and  operose 


and  sin  ic  — 


sin  a = 


■fcVfr  (¥~a)  • 0-6)  • Qs-c) 

, O-c). 

_ ^ 


E = 55°-771 191  = 55  46  16  18. 

c = 82Q-819206  = 82  49  9 8. 


Solution 


24  ANALYTICAL  PLANE  TRIGONOMETRY 


Solution  to  the  same  by  R Adrain,  , 

Let  abc  be  the  triangle,  having  the  angle 
abc  double  the  angle  a,  produce  ab  to  d,  mak- 
ing bd  = bc,  and  join  cd  ; and  the  triangles 
cbd,  acd  are  evidently  isosceles  and  equian- 
gular ; therefore  bd  or  bc  is  to  cd  or  ac  as  ac 
to  ad.  Now  let  ab  — x,  bc  = x — 1,  ac  = x -f-  1 , then 
ad  — 2.x — 1,  and  the  preceding  stating  becomes  x — 1: 
x + \ : : x + l : 2x—  1,  which  by  multiplying  extremes  and 
means  gives  2x2  — 3x  + l — x2  -j-  2.r  + 1,  and  by  -ubtrac- 
tion  x2  — ox,  or  dividing  by  x,  simply  x = 5,  hence  the  sides 
are  4,  5,  6. 

The  same  conclusion  is  also  readily  obtained  without  the 
use  of  algebra 

Ex.  7.  Demonstrate  that  sin  18°  = cos  72°  is  = Ar. 
( — 1 -f^/5),  and  sin  54°  = cos  36°  is  — ar  (1  + y/  5). 

Ex.  8.  Demonstrate  that  the  sum  of  the  sines  of  two  arcs 
which  together  make  60°,  is  equal  to  the  sine  of  an  arc 
which  is  greater  than  60,  by  either  of  the  two  arcs  : Ex.  gr. 
sin  3'  + sin  59°  57'  = sin  60°  3’  ; and  thus  that  the  tables  may 
be  continued  by  addition  only. 

Ex  9 Show  the  truth  of  the  following  proportion  : As 
the  sine  of  half  the  difference  of  two  arcs,  which  together 
make  60°,  or  90°,  respectively,  is  to  the  difference  of  their 
sines  ; so  is  1 to  ^/2,  or  ^/3,  respectively. 

Ex.  10.  Demonstrate  that  the  sum  of  the  square  of  the 
sine  and  versed  sine  of  an  arc,  is  equal  to  the  square  of  dou- 
ble the  sine  of  half  the  arc. 

Ex.  11.  Demonstrate  that  the  sine  of  an  arc  is  a mean 
proportional  between  half  the  radius  and  the  versed  sine  of 
double  the  arc. 

Ex.  12.  Show  that  the  secant  of  an  arc  is  equal  to  the 
sum  of  its  tangent  and  th(e  tangent  of  half  its  complement. 

Ex.  13.  Prove  that,  in  any  plane  triangle,  the  base  is  to 
the  difference  of  the  other  two  sides,  as  the  sine  of  half  the 
sum  of  the  angles  at  the  base,  to  the  sine  of  half  their  diffe- 
rence : also,  that  the  base  is  to  the  sum  of  the  other  two  sides, 
as  the  cosine  of  half  the  sum  of  the  angles  at  the  base,  to  the 
cosine  of  half  their  difference. 


Ex. 


ANALYTICAL  PLANE  TRIGONOMETRY.  25 

Ex.  14.  How  must  three  trees,  a,  b,  c,  be  planted,  sa 
that  the  angle  at  a may  be  double  the  angle  at  b,  the  angle 
at  b double  that  at  c ; and  so  that  a line  of  400  yards  may 
just  go  round  them  ? 

Ex.  15.  In  a certain  triangle,  the  sines  of  the  three  an- 
gles are  as  the  numbers  17,  15,  and  8,  and  the  perimeter  is 
160.  What  are  the  sides  and  angles  ? 

Ex.  16.  The  logarithms  of  two  sides  of  a triangle  are 
2-2407293'  and  2-5378191,  and  the  included  angle,  is  37®  20'. 
It  is  required  to  determine  the  other  angles,  without  first 
finding  any  of  the  sides  ? 

Ex.  17.  The  sides  of  a triangle  are  to  each  other  as  the 
fractions  a,  | : what  are  the  angles  ? 

Ex.  18.  Show  that  the  secant  of  60°,  is  double  the  tan- 
gent of  45°,  and  that  the  secant  of  45°  is  a mean  proportional 
between  the  tangent  of  45°  and  the  secant  of  60®. 

Ex.  19.  Demonstrate  that '4  times  the  rectangle  of  the 
sines  of  two  arcs,  is  equal  to  the  difference  of  the  squares  of 
the  chords  of  the  sum  and  difference  of  those  arcs. 

Ex.  20.  Convert  the  equations  marked  xxxiv  into  their 
equivalent  logarithmic  expressions  ; and  by  means  of  them 
and  equa  iv,  find  the  angles  of  a triangle  whose  sides  are  6, 
f,  and  7. 

SPHERICAL  TRIGONOMETRY- 


SECTION  I. 

General  Properties  of  Spherical  Triangles. 

Art.  I.  Def.  1.  Any  portion  of  a spherical  surface  bounded 
by  three  arcs  of  great  circles  is  called  a Spherical  Triangle. 

Def.  2.  Spherical  Trigonometry  is  the  art  of  computing 
the  measures  of  the  sides  and  angles  of  spherical  triangles. 
Vpfc.  IK  5 Def. 


26  SPHERICAL  TRIGONOMETRY. 

Def.  3.  A right  angled  spherical  triangle  has  one  right 
angle  : the  sides  about  the  right  angle  are  called  tegs  ; the 
side  opposite  to  the  right  angle  is  called  the  hypothenuse 

■ Def.  4.  A quadrantal  spherical  triangle  has  one  side  equal 
to  90°  or  a quarter  of  a great  circle. 

Def.  5.  Two  arcs  or  angles,  when  compared  together,  are 
said  to  be  alike,  or  of  the  same  affection,  when  both  are  less 
than  90°,  or  both  are  greater  than  90°.  But  when  one  is 
greater  and  the  other  less  than  90°,  they  are  said  to  be  unlike, 
or  of  different  affections. 

Aut.  2.  The  small  circles  of  the  sphere  do  not  fall  under 
consideration  in  Spherical  Trigonometry  ; but  such  only  as 
have  the  same  centre  with  the  sphere  itself.  ' And  hence  it  is 
that  spherical  trigonometry  is  of  so  much  use  in  Practical 
Astronomy,  the  apparent  heavens  assuming  the  shape  of  a 
concave  sphere,  whose  centre  is  the  same  as  the  centre  of  the 
earth. 

3.  Every  spherical  triangle  has  three  sides,  and  three  an- 
gles : and  il  any  three  of  these  six  parts,  he  given,  the  re- 
maining three  may  be. found,  by  some  of  the  rules  which 
will  be  investigated  in  this  chapter. 

4 In  plane  trigonometry,  the  knowledge  of  the  three  an- 
gles is  not  sufficient  for  ascertaining  the  sides  : for  in  that 
case  the  relations  only  of  the  three  sides  can  be  obtained,  and 
not  their  absolute  values  : whereas,  in  spherical  trigonome- 
try, where  the  sides  are  circular  arcs,  whose  values  depend 
on  their  proportion  to  the  whole  circle,  that  is,  on  the  num- 
ber of  degrees  they  contain,  the  side-  may  always  be  deter- 
mined when  the  three  angles  are  known.  Other  remarkable 
differences  between  plane  and  spherical  triangles  are,  1st. 
That  in  the  former,  two  angles  always  determine  the  third  ; 
while  in  the  latter  they  never  do.  2dly.  The  surface  of  a 
plane  triangle  cannot  be  determined  from  a knowledge  of  the 
angles  alone  ; -while  that  of  a spherical  triangle  always  can. 

5.  The  sides  of  a spherical  triangle  are  all  arcs  of  great 
circles,  which,  by  their  intersection  on  the  surface  of  the 
sphere,  constitute  that  triangle. 

6.  The  angle  which  is  contained  between  the  arcs  of  two 
great  circles,  intersecting  each  other  on  the  surface  of  the 
sphere,  is  called  a spherical  angle  ; and  its  measure  is  the  same 
as  the  measure  ot  the  plane  angle  which  is  formed  by  two 
lines  issuing  from  the  satne  point  of,  and  perpendicular  to, 
the  common  section  of  the  planes  wffiich  determine  the  con- 
taining 


SPHERICAL  TRIGONOMETRY. 


27 


taining  sides  : that  is  to  say,  it  is  the  same  as  the  angle  made 
by  those  planes.  Or,  it  is  equal  to  the  plane  angle,  formed 
by  the  tangents  to  those  arcs  at  their  point  of  intersection. 

7.  Hence  it  follows,  that  the  surface 
of  a spherical  triangle  bac,  and  the 
three  planes  which  determine  it  form 
a kind  of  triangular  pyramid,  bcga 
of  which  the  vertex  g is  at  the  centre 
of  the  sphere;  the  base  abc  a portion 
of  the  spherical  surface,  and  the  faces 
agc.  aCb,  bgc,  sectors  of  the  great 
circles  whose  intersections  determine 
the  sides  of  the  triangle. 

Def.  6.  A line  perpendicular  to  the  plane  of  a great  circle, 
passing  through  the  centre  of  the  sphere,  and  terminated  by 
two  points,  diametrically  opposite,  at  its  surface,  is  called  the 
axis  of  suth  a circle  ; and  the  extremities  of  the  axis,  or  the 
points  where  it  meets  the  surface,  are  called  the  poles  of  that 
circle.  Thus,  pgp'  is  the  axis,  and  p,  p',  are  the  poles,  of  the 
great  circle  cnd. 

If  vve  conceive  any  number  of  less  circles,  each  parallel  to 
the  said  great  circle,  this  axis  will  be  perpendicular  to  them 
likewise  ; and  the  points  p,  p'  will  be  their  poles  also. 

8.  Hence,  each  pole  of  a great  circle  is  90°  distant  from 
every  point  in  its  circumference  ; and  all  the  arcs  drawn  from 
either  pole  of  a little  circle  to  its  circumference,  are  equal  to 
each  other. 

9.  It  likewise  follows,  that  all  the  arcs  of  great  circles  drawn 
through  the  poles  of  another  great  circle,  are  perpendicular 
to  it  : for  since  they  are  great  circles  by  the  supposition, 
they  all  pass  through  the  centre  of  the  sphere,  and  conse- 
quently through  the  axis  of  the  said  circle.  The  same  thing 
may  be  affirmed  with  regard  to  small  circles. 

10.  Hence,  in  order  to  find  the  poles  of  any  circle,  it  is 
merely  necessary  to  describe,  upon  the  surface  of  the  sphere, 
two  great  circles  perpendicular  to  the  place  of  the  former  ; 
the  points  where  these  circles  intersect  each  other  will  be  the 
poles  required. 

11.  It  may  be  inferred  also,  from  the  preceding,  that  if  it 
were  proposed  to  draw,  from  any  point  assumed  on  thj  sur- 
face of  the  sphere,  an  arc  of  a circle  which  may  measure  the 
shortest  distance  from  that  point,  to  the  circumference  of 
any  given  circle  ; this  arc  must  be  so  described,  that  its  pro- 
longation may  pass  through  the  poles  of  the  given  circle. 
And  conversely,  if  an  arc  pass  through  the  poles  of  a given 

circle, 


28 


SPHERICAL  TRIGONOMETRY, 


circle,  it  will  measure  the  shortest  distance  from  any  assumed 
point  to  the* circumference  of  that  circle. 

12.  Hence  again,  if  upon  the  sides,  ac  and  bc,  (produced 
if  necessary)  of  a spherical  triangle  bca,  we  take  the  arcs,  ex, 
cm,  each  equal  90°,  and  through  the  radii  gn,  cm  (figure  to 
art.  7)  draw  the  plane  ngm,  it  is  manifest  that  the  point  c 
will  be  the  pole  of  a circle  coinciding  with  the  plane  ngm  : 
«o  that,  as  the  lines  gm,  gn,  are  both  perpendicular  to  the 
common  section  gc,  of  the  planes  agc,  bgc,  they  measure,  by 
their  inclination  the  angle  of  these  planes  ; or  the  arc  nm 
measures  that  angle,  and  consequently  the  spherical  angle  bca. 

13.  It  is  also  evident  that  every  arc  of  a little  circle,  des- 
cribed from  the  pole  c as  centre,  and  containing  the  same 
number  of  degrees  as  the  arc  mn,  is  equally  proper  for  mea- 
suring the  angle  bca  ; though  it  is  customary  to  use  only  arcs 
of  great  circles  for  this  purpose. 

14.  Lastly,  we  infer,  that  if  a spherical  angle  be  a right 
angle,  the  arcs  of  the  great  circles  which  form  it,  will  pass 
mutually  through  the  poles  of  each  other  : and  that,  if  the 
planes  of  two  great  circles  contain  each  the  axis  of  the  other, 
or  pass  through  the  poles  of  each  other,  the  angle  which  they 
include  is  a right  angle. 

These  obvious  truths  being  premised  and  comprehended, 
the  student  may  pass  to  the  consideration  of  the  following 
theorems. 

THEOREM  I. 

Any  Two  Sides  of  a Spherical  Triangle  are  together  Greater 
than  the  Third. 

This  proposition  is  a necessary  consequence  of  the  truth, 
that  the  shortest  distance  between  any  two  points,  measured 
on  the  surface  of  the  sphere,  is  the  arc  of  a great  circle  pass- 
ing through  these  points. 

THEOREM  II. 

The  Sum  of  the  Three  Sides  of  any  Spherical  Triangle  is 
Less  than  360  degrees. 

For,  let  the  sides  ac,  bc,  (fig.  to  art.  7)  containing  any 
angle  a,  be  produced  till  they  meet  again  in  n : then  will  the 
arcs  dac,  dec,  be  each  180°,  because  all  great  circles  cut  each 
other  into  two  equal  parts  : consequently  dac  + dbc  = 360°. 
But  (theorem  1)  ba  and  db  are  together  greater  than  the 

third 


SPHERICAL  TRIGONOMETRY. 


29 

third  side  ab  of  the  triangle  dab  ; and  therefore,  since  ca  -f- 
gb  -f-  da  -(-  db  = 360°,  the  sum  ca  -j-  cb  + ab  is  less  than 
360°.  Q.  e.  d. 

THEOREM  HI. 

The  Sum  of  the  Three  Angles  of  any  Spherical  Triangle  is 

always  Greater  than  Two  Right  Angles,  but  less  than  Six. 

1.  The  first  part  of  this  theorem  is  demonstrated  in  cor.  2 
of  the.  iv.  following. 

2.  The  angle  of  inclination  of  no  two  of  the  planes  can  be 

so  great  as  two  right  angles  ; because,  in  that  case,  the  two 
planes  would  become  but  one  continued  plane,  and  the  arcs, 
instead  of  being  arcs  of  distinct  circles,  would  be  joint  arcs  of 
one  and  the  same  circle.  Therefore,  each  of  the  three  sphe- 
rical angles  must  be  less  than  two  right  angles  ; and  conse- 
quently their  sum  less  than  six  right  angles.  q e.  d. 

Cor.  1.  Hence  it  follows,  that  a spherical  triangle  may 
have  all  its  angles  either  right  or  obtuse  ; and  therefore  the 
knowledge  of  any  two  right  angles  is  not  sufficient  for  the  de- 
termination of  the  third. 

Cor.  2.  If  the  three  angles  of  a spherical  triangle  be  right 
or  obtuse,  the  three  sides  are  likewise  each  equal  to,  or  greater 
than  9U°  : and,  if  each  of  the  angles  be  acute,  each  of  the  sides 
is  also  less  than  90  ; and  conversely. 

Scholium.  From  the  preceding  theorem  the  student  may 
clearly  perceive  what  is  the  essential  difference  between  plane 
and  spherical  triangles,  and  how  absurd  it  would  be  to  apply 
the  rules  of  plane  trigonometry  to  the  solution  of  cases  in 
spherical  trigonometry.  Yet,  though  the  difference  hotween 
the  two  kinds  of  triangles  be  really  so  great,  still  there  are 
various  properties  which  are  common  to  both,  and  which  may 
be  demonstrated  exactly  in  the  same  manner.  Thus,  for  ex- 
ample, it  might  be  demonstrated  here,  (as  well  as  with  regard 
to  plane  triangles  in  the  elements  of  Geometry,  vol  1)  that 
two  spherical  triangles  are  equal  to  each  other,  1st.  When  the 
three  sides  of  the  one  are  respectively  equal  to  the  three  sides 
of  the  other.  2dly.  When  each  of  them  has  an  equal  angle 
contained  between  equal  sides  : and,  3dly.  When  they  have 
each  two  equal  angles  at  the  extremities  of  equal  bases.  It 
might  also  be  shown,  that  a spherical  triangle  is  equilateral, 
isosceles,  or  scalene  according  as  it  hath  three  equal,  two 
equal,  or  three  unequal  angles  : and  again,  that  the  greatest 
side  is  always  opposite  to  the  greatest  angle,  and  the  least  side 

to 


30  SPHERICAL  TRIGONOMETRY. 

to  the  least  angle.  But  the  brevity  that  our  plan  requires, 
compels  us  merely  to  mention  these  particulars  It  may  be 
added,  however,  that  a spherical  triangle  may  be  at  once 
right-angled  and  equilateral ; which  can  never  be  the  case  with 
a plane  triangle. 

THEOREM  IV. 

If  from  the  Angles  of  a Spherical  Triangle,  as  Poles,  there 
be  described,  on  the  Surface  of  the  Sphere,  Three  Arcs  of 
Great  Circles,  which  by  their  Intersections  'form  another 
Spherical  Triangle  ; Each  Side  of  this  New  Triangle  will 
be  the  Supplement  to  the  Measure  of  the  Angle  which  is  at 
its  Pole,  and  the  Measure  of  each  of  its  Angles  the  Supple- 
ment to  that  Side  of  the  Primitive  Triangle  to  which  it  is 
Opposite. 

From  b,  a,  and  c,  as  poles,  let  the 
arcs  df,  de,  fe,  be  described,  and  by 
their  intersections  form  another  spheri- 
cal triangle  def  ; either  side,  as  de,  of 
this  triangle,  is  the  supplement  of  the 
measure  of  the  angle  a at  its  pole  ; and 
either  angle,  as  d,  has  for  its  measure 
the  supplement  of  the  side  ab. 

Let  the  sides  ab,  ac,  bc,  of  the  primitive  triangle,  be  pro- 
duced till  they  meet  those  of  the  triangle  def.  in  the  points 
i,  l,  m,  n,  g,  k : then,  since  the  point  a is  the  pole  of  the  arc 
djle,  the  distance  of  the  points  a and  e (measured  on  an  arc 
of  a great  circle)  will  be  90c  ; also,  since  c is  the  pole  of  the 
arc  ef,  the  points  c and  e will  be  90°  distant  : consequently 
(art.  8)  the  point  e is  the  pole  of  the  arc  ac.  In  like  manner 
it  may  be  shown,  that  r is  the  pole  of  bc,  and  d that  of  ab. 

This  being  premised,  we  shall  have  dl  = 90°,  and  ie  = 90c 
whence  dl  + ie  = dl  -j-  el  -j-  il  = de  + jl  = 180°. 
Therefore  de  = 180°— il  : that  is,  since  il  is  the  measure 
of  the  angle  bac,  the  arc  df.  is  = the  supplement  of  that 
measure.  Thus  also  may  it  he  demonstrated  that  ef  is  equal 
the  supplement  to  mn,  the  measure  of  the  angle  bca,  and 
that  df  is  equal  the  supplement  to  gk,  the  measure  of  the 
angle  abc  : which  constitutes  the  first  part  of  the  proposition. 

2dly.  The  respective  measures  of  the  angles  of  the  triangle 
def  are  supplemental  to  the  opposite  sides  of  the  triangles 
abc.  For,  since  the  arcs  al  and  bg  are  each  90°,  therefore 

fe 


1? 


SPHERICAL  TRIGONOMETRY. 


o 1 

o 1 


ia  At  + eg  = gl  4*  ab  = 180°  ; whence  gl  ==  180®  — ab  ; 
that  is,  the  measure  of  the  angle  d is  equal  to  the  supplement 
to  ab.  So  likewise  may  it  be  shown  that  ac,bc,  are  equal  to 
the  supplements  to  the  measures  of  the  respectively  opposite 
angles  e and  f.  Consequently,  the  measures  of  the  angles 
of  the  triangle  def  are 'supplemental  to  the  several  opposite 
sides  of  the  triangle  abc.  q.  e.  d. 

Cor.  1.  Hence  these  two  triangles  are  called  supplemental 
or  polar  triangles. 

Cor.  2.  Since  the  three  sides  de,  ef,  df,  are  supplements 
to  the  measures  of  the  three  angles  a,  b,  c ; it  results  that 
de  + ef  4-  DF  + A -f-  b -f-  c = 3X  1S0°=540Q  But  (th  2), 
de  4*  ef  4"  DF  < 3 0°:  consequently  a 4"  B 4 c > 180°. 
Thus  the  first  part  of  theorem  3 is  very  compendiously  de- 
monstrated 

Cor.  3 This  theorem  suggests  mutations  that  are  some- 
times of  use  in  computation. — Thus,  if  three  angles  of  a 
spherical  triangle  aie  given,  to  find  the  sides  : the  student 
may  subtract  each  of  the  angles  from  180°,  and  the  three  re- 
mainders will  be  the  three  sides  of  a new  triangle  ; the  angles 
of  this  n<-w  triangle  being  found,  if  their  measures  be  each  ta- 
ken from  180°,  the  three  remainders  will  be  the  respective 
sides  of  the  primitive  triangle,  whose  angles  were  given 

Scholium.  The  invention  of  the  preceding  theorem  is  due 
to  Philip  Langsberg  Vide,  Simon  Steven,  liv  3,  de  la  Cos- 
mographie,  prop  31  and  Alb.  Girard  in  loc.  It  is  often  how- 
ever treated  very  loosely  by  authors  on  trigonometry  : some 
of  them  speaking  of  sides  as  the  supplements  of  angles,  and 
scarcely  any  of  them  remarking  which  of  the  several  triangles 
formed  by  the  intersection  of  the  arcs  de,  ef.  df,  is  the  ope 
in  question.  Besides  the  triangle  def,  three  others  may  be 
formed  by  the  intersection  of  the  semi- 
circles, and  if  the  whole  circles  be  consi- 
dered, there  will  be  seven  other  triangles 
formed  But  the  proposition  only  obtains 
with  regard  to  the  central  triangle  (of 
each  hemisphere),  which  is  distinguished 
from  the  three  others  in  this,  that  the 
two  angles  a and  f are  situated  on  the 
same  side  of  bc,  the  two  b and  e on  the 
the  two  c and  d on  the  same  side  of  ab. 


THEOREM  V. 


In  Every  Spherical  Triangle,  the  following  proportion  obtains, 
viz,  As  Four  Right  Angles  (or  360°)  to  the  surface  of  a 

Hemisphere 


32 


SPHERICAL  TRIGONOMETRY. 


Hemisphere  ; or,  as  Two  Right  Angles' (or  180°)  to  a Great 
Circle  of  the  Sphere  ; so  is  the  Excess  of  the  three  angle* 
of  the  Triangle  above  Two  Right  Angles,  to  the  Area  of  the 
triangle. 

Let  abc  be  the  spherical  triangle.  Com- 
plete one  of  itg  sides  as  bc  into  the  circle 
bcef,  which  may  be  supposed  to  bound 
the  upper  hemisphere.  Prolong  also,  at 
both  ends,  the  two  sides  ab.  ac,  until  they 
form  semicircles  estimated  from  each  an- 
gle, that  is,  until  bae  = abd  = caf  = 
act/—  180°.  Then  will  cbf=  180°=bff.  ; 
and  consequently  the  triangle  aef,  on  the  anterior  hemisphere 
will  be  equal  to  the  triangle  bcd  on  the  opposite  hemisphere. 
Putting  m,  in'  to  represent  the  surface  of  these  triangles,  p 
for  that  of  the  triangle'  baf,  q for  that  of  cae,  and  a for  that 
of  the  proposed  triangle  abc.  Then  a and  m together  (or  their 
equal  a and  m together)  make  up  the  surface  of  a spheric  lune 
comprehended  between  the  two  semicircles,  acd,  abd.  inclin- 
ed in  the  angle  a : a and  p together,  make  up  the  lune  in- 
cluded between  the  semicircles  caf,  cbf,  making  the  angle  c : 
a and  q together  make  up  the  spheric  lune  included  between 
the  semicircles  bce,  bae  making  the  angle  b.  And  the  sur- 
face of  each  of  these  lunes,  is  to  that  of  the  hemisphere,  as  the 
angle  made  hy  the  comprehending  semicircles,  to  two  right 
angles.  Therefore,  putting  is  for  the  surface  of  the  hemi- 
sphere, we  have 

180°  : a : : ^s  : a -f-  m. 

180°  : b : : is  : a + q- 

180°  : c : : is  : a + p. 

Whence,  180°  : a+b+c  : : is  : 3a+m  + p + 7 = 2a  + is  ; 

and  consequently,  by  division  of  proportion. 

as  1 80°  : a + b + c — 1 80°  : : is  : 2a  -f-  is  — - 4s  = 2a  ; 

a+ b+c-  1 80° 

or,  180°  : a b -{-  c — 180°  : : is  : a — £s. 360° 

e.  d.* 

Cor.  1.  Hence  the  excess  of  the  three  angles  of  any  spheri- 
cal triangle  above  two  right  angles,  termed  technically  the 


* This  determination  of  the  area  of  a spherical  triangle  is  due  to 
Albert  Girard  (who  died  about  1633).  But  the  demonstration  now  com- 
monly given  of  the  rule  was  first  published  by  Dr.  YV  allis.  It  was  con- 
sidered as  a mere  speculative  truth,  until  General  Hoy,  in  1787-  em- 
ployed it  very  judiciously  in  the  great  Trigonometrical  Survey,  to  cor- 
rect the  errors  of  spherical  angles.  See  Phil.  Trans,  vol.  80,  and  the 
next  chapter  of  this  volume- 

spherical 


SPHERICAL  TRIGONOMETRY. 


33 


spherical  excess,  furnishes  a correct  measure  of  the  surface 
of  that  triangle. 

Cor.  2.  If  re  = 3-141593,  and  d the  diameter  of  the 
sphere,  then  is  re d*.  A~^~B  18  J = the  area  of  the  spherical 


triangle. 

Cor.  3.  Since  the  length  of  the  radius,  in  any  circle,  is 
equal  to  the  length  of  57-2957795  degrees,  measured  on  the 
circumference  of  that  circle  ; if  the  spherical  excess  be  mul- 
tiplied by  57-297795,  the  product  will  express  the  surface  of 
the  triangle  in  square  degrees. 

Gor.  4.  When  a = 0,  then  a -f-  b -f-  c — 180°  : and  when 
a = J-s,  then  a b + c = 540°.  Consequently  the  sum  of 
the  three  angles  of  a spherical  triangle,  is  always  between  2 
and  6 right  angles  : which  is*  another  confirmation  of  th.  3. 

Cor.  5.  When  two  of  the  angles  of  a spherical  triangle 
are  right  angles,  the  surface  of  the  triangle  varies  with  its 
third  angle.  And  when  a spherical  triangle  has  three  right 
angles  its  surface  is  one  eighth  of  the  surface  of  the  sphere. 

Remark.  Some  of  the  uses  of  the  spherical  excess,  in  the 
more  extensive  geodesic  operations,  will  be  shown  in  the  fol- 
lowing chapter.  The  mode,  of  finding  it,  and  thence  the  area 
when  the  three  angles  of  a spherical  triangle  are  given,  is  ob- 
vious enough  ; but  it  is  often  requisite  to  ascertain  it  by  means 
of  other  data,  as  when  two  sides  and  the  included  angle  are 
given,  pr  when  all  the  three  sides  are  given.  In  the  former 
case,  let  a and  b be  the  two  sides,  c the  included  angle,  and 

, COt  Cat  i b 4-  COS  C 

e the  spherical  excess  : then  is  cot  4 e = ; 

1 sin  c 

When  the  three  sides  a,  b , c,  are  given,  the  spherical  excess 
may  be  found  by  the  following  very  elegant  theorem,  discover- 
ed by  Simon  Lhuillier  : 


tani  e — v/  (tan  - 


a-f-£-f-c 


a+b—e  a-  A-fc 
tan . tan  — - — . tan 


a-\-b  +c 


4 


The  investigation  of  these  theorems  would  occupy  more  space 
than  can  be  allotted  to  them  in  the  present  volume. 


THEOREM  VI. 


In  every  Spherical  Polygon,  or  surface  included  by  any  num- 
ber of  intersecting  great  circles,  the  subjoined  proportion 
obtains,  viz.  As  Four  Right  Angles,  or  360°,  to  the  Surface 
of  a Hemisphere  ; or,  as  Two  Right  Angles,  or  1S0°,  to  a 
Great  Circle  of  the  Sphere  ; so  is  the  Excess  of  the  Sum 
of  the  Angles  above  the  Product  of  180°  and  fwo  Less 
than  the  number  of  Angles  of  the  spherical  polygon,  to 
its  Area. 


Yon.  Ih 


For, 


6 


34 


SPHERICAL  TRIGONOMETRY, 


For,  if  the  polygon  be  supposed  to  be  divided  into  as  many 
triangles  as  it  has  sides,  by  great  circles  drawn  from  all  the 
angles  through  any  point  within  it,  forming  at  that  point  the 
vertical  angles  of  all  the  triangles.  Then,  by  th.  5,  it  will  be 
as  360°  : is  : : a-Rb-Rc — 180°  : its  area.  Therefore,  put- 
ting p for  the  sum  of  all  the  angles  of  the  polygon,  n for  their 
number,  and  v for  the  sum  of  all  the  vertical  angles  of  its 
constituent  triangles,  it  will  be,  by  composition, 
as  360°  : is  : : p + v — 180°  n : surface  of  the  polygon. 
But  v is  manifestly  equal  to  360°  or  180°  X 2.  Therefore, 

as  360°  : is  : : p — (n — 2)  180°  : is.  F~~^~ 18U  -,  the  area 

of  the  polygon,  q.  e.  d. 

Cor.  1.  If  ir  and  d represent  the  same  quantities  as  in 
theor.  5 cor.  2,  then  the  surface  of  the  polygon  will  be  ex- 
pressed by  vrd*. 

Cor.  2.  If  r°  = 57-2957795,  then  will  the  surface  of  the 
polygon  in  square  degrees  be  = r°.  (p  — (n  — £)  180°). 

Cor.  3.  When  the  surface  of  the  polygon  is  O,  then  p = 
(n  — 2)  180°  ; and  when  it  is  a maximum,  that  is,  when  it  is 
equal  to  the  surface  of  the  hemisphere,  then  p = (n — 2)  180° 
360°  = n . 180°  ~.  Consequently  p,  the  sum  of  all  the  an- 
gles of  any  spheric  polygon,  is  always  less  than  2n  right  angles, 
but  greater  than  (2n— 4)  right  angles  n,  denoting  the  num- 
ber of  angles  of  the  polygon, 

GENERAL  SCHOLIUM; 


On  the  Nature  and  Measure  of  Solid  Angles- 

A Solid  Angle  is  defined  by  Euclid,  that  which  is  made  by 
the  meeting  of  more  than  two  plane  angles,  which  are  not  in 
the  same  plane,  in  one  point. 

Others  define  it  the  angular  space  comprized  between 
several  planes  meeting  in  one  point. 

It  may  be  defined  still  more  generally,  the  angular  space 
included  between  several  plane  surfaces  or  one  or  more  curv- 
ed surfaces,  meeting  in  the  point  which  forms  the  summit  of 
the  angle. 

According  to  this  definition,  solid  angles  bear  just  the  same 
relation  to  the  surfaces  which  comprize  them,  as  plane  angles 
do  to  the  lines  by  which  they  are  included  : so  that,  as  in  the 
latter,  it  is  not  the  magnitude  of  the  lines,  but  their  mutual 
inclination,  which  determines  the  angle  ; just  so,  in  the  former 

it 


SPHERICAL  TRIGOMETRY. 


35 

it  is  not  the  magnitude  of  the  planes,  but  their  mutual  inclina- 
tions which  determine  the  angles.  And  hence  all  those  ge- 
ometers, from  the  time  of  Euclid  down  to  the  present  period, 
who  have  confined  their  attention  principally  to  the  magnitude 
of  the  plane  angles  instead  of  their  relative  positions,  have 
never  been  able  to  develope  the  properties  of  this  class  of 
geometrical  quantities  ; but  have  affirmed  that  no  solid  angle 
can  be  said  to  be  the  half  of  the  double  of  another,  and  have 
spoken  of  the  bisection  and  trisection  of  solid  angles,  even  in 
the  simplest  cases,  as  impossible  problems. 

But  all  this  supposed  difficulty  vanishes,  and  the  doctrine  of 
solid  angles  becomes  simple,  satisfactory,  and  universal  in  its 
application,  by  assuming  spherical  surfaces  for  their  measure  ; 
just  as  circular  arcs  are  assumed  for  the  measures  of  plane 
angles*.  Imagine,  that  from  the  summit  of  a solid  angle, 
(formed  by  the  meeting  of  three  planes)  as  a centre,  any  sphere 
be  described,  and  that  those  planes  are  produced  till  they  cut 
the  surface  of  the  sphere  ; then  will  the  surface  of  the  spheri- 
cal triangle,  included  between  those  planes  be  a proper  mea- 
sure of  the  solid  angle  made  by  the  planes  at  their  common 
point  of  meeting  ; for  no  change,  can  be  conceived  in  the  rela- 
tive position  of  those  planes,  that  is  in  the  magnitude  of  the 
solid  angle,  without  a corresponding  and  proportional  mutation 
in  the  surface  of  the  spherical  triangle.  If,  in  like  manner, 
the  three  or  more  surfaces  tyhich  by  their  meeting  constitute 
another  solid  angle,  be  produced  till  they  cut  the  surface  of 
the  same  or  an  equal  sphere,  whose  centre  coincides  with  the 
summit  of  the  angle  ; the  surface  of  the  spheric  triangle  or 
polygon,  included  between  the  planes  which  determine  the 


* it  may  be  proper  to  anticipate  here  the  only  objection  which  can 
be  ma  le  to  this  assumption  ; wh  ch  Is  founded  on  the  principle,  that 
quantities  should  always  be  measured  by  quantities  of  the  same  kind.  But 
this,  often  and  pos  .ively  as  a is  affi  med,  is  by  no  means  necessary  ; 
nor  in  many  cases  is 't  possible.  To  measure  is  to  compufirmathemati- 
caiiy  : and  if  by  comparing  two  quantites,  whose  ratio  we  know  or  can. 
ascertain,  witu  two  other  quant  t:es  whose  ratio  we  wish  to  know,  the 
point  in  question  becomes  determined:  it  signifies  not  at  alt  wnether 
the  magnitudes  wnich  constitute  one  ratio,  are  nke  or  uni  ke  the  mag- 
nitudes which  constitute  the  other  rati).  It  is  thus  that  mathemati- 
cians, with  perfect  safety  and  correctness,  make  use  of  sp  ice  as  a mea- 
sure of  velocity,  mass  as  a measure  of  inertia,  muss  and  velocity  con- 
jointly as  a measure  of  force,  sp  tce  as  a measure  of  time,  w ight  as  a 
measure  of  density,  expansion  as  a measure  of  heat,  a certain  function 
of  planetary  velocity  as  a measure  of  distance  from  the  central  bony, 
arcs  of  the  same  circle  as  measures  of  plane  angles  ; and  it  is  in  con- 
formity with  this  general  procedure  that  we  adopt  surfaces,  of  the  same 
sphere,  as  measures  of  solid  angles. 


36 


SPHERICAL  TRIGONOMETRY. 


angle,  will  be  a correct  measure  of  that  angle.  And  the  ratio 
which  subsists  between  the  areas  of  the  spheric  triangles  po- 
lygons, or  other  surfaces  thus  formed,  will  be  accurately  the 
ratio  which  subsists  between  the  solid  angles,  constituted  bi' 
the  meeting  of  the  several  planes  or  surfaces,  at  the  centre  of 
the  sphere. 

Hence,  the  comparison  of  solid  angles  becomes  a matter  of 
great  ease  and  simplicity  : for.  since  the  areas  of  spherical 
triangles  are  measured  by  the  excess  of  the  sums  of  their  an- 
gles each  above  two  right  angles  (th.  5)  ; and  the  areas  ot 
spherical  polygons  of  n sides,  by  the  excess  of  the  sum  of 
their  angles  abeve  2 n — 4 right  angles  (th.  6)  ; it  follows,  that 
the  magnitude  of  a trilateral  solid  angle,  will  be  measured  by 
the  excess  of  the  sum  of  the  three  angles,  made  respectively 
by  its  bounding  planes,  above  2 right  angles  ; and  the  magni- 
tudes of  solid  angles  formed  by  n bounding  planes,  by  the  ex- 
cess of  the  sum  of  the  angles  of  inclination  of  the  several 
planes  above  2 n — 4 right  angles. 

As  to  solid  angles  limited  by  curve  surfaces,  such  as  the  an- 
gles at  the  vertices  of  cones  ; they  will  manifestly  be  measur- 
ed by  the  spheric  surfaces  cut  oil'  by  the  prolongation  of  their 
bounding  surfaces,  in  the  same  manner  as  angles  determined 
by  planes  are  measured  by  the  triangles  or  polygons,  they 
mark  out  upon  the  same,  or  an  equal  sphere.  In  all  cases, 
the  maximum  limit  of  solid  angles,  will  be  the  plane  towards 
which  the  various  planes  determining  such  angles  approach, 
as  they  diverge  further  from  each  other  about  the  same  sum- 
mit : just  as  a right  line  is  the  maximum  limit  of  plane  angles, 
being  formed  by  the  two  bounding  lines  when  they  make  an 
angle  of  180°.  The  maximum  limit  of  solid  angles  is  meas- 
ured by  the  surface  of  a hemisphere,  in  like  manner  as  the 
maximum  limit  of  plane  angles  is  measured  by  the  arc  of  a 
semicircle.  The  solid  right  angle  (either  angle,  for  example 
of  a cube)  isj  (=-p)  of  the  maximum  solid  angle  : while  the 
plane  right  angle  is  half,  the  maximum  plane  angle. 

The  analogy  between  plane  and  solid  angles  being  thus  tra- 
ced, we  may  proceed  to  exemplify  this  theory  by  a few  instan- 
ces ; assuming  1000  as  the  numeral  measure  of  the  maximum 
solid  angle  = 4 times  90°  solid  = 360°  solid. 

1.  The  solid  angles  of  right  prisms  are  compared  with  great 
facility.  For,  of  the  three  angles  made  by  the  three  planes 
which,  by  their  meeting,  constitute  every  such  solid  angle, 
two  are  right  angles  : and  the  third  is  the  same  as  the  corres- 
ponding plane  angle  of  the  polygonal  base  ; on  which,  there- 
fore, the  measure  of  the  solid  angle  depends.  Thus,  with 

respect  - 


SPHERICAL  TRIGONOMETRY. 


37 


respect  to  the  right  prism  with  an  equilateral  triangular  base, 
each  solid  angle  is  formed  by  planes  which  respectively  make 
angles  of  90°,  90°,  and  60°.  Consequently  90°  -{-  90°  + 60°- 
180°  = 60°,  is  the  measure  of  such  angle,  compared  with  360° 
the  maximum  angle.  It  is  therefore,  one-sixth  of  the  maxi- 
mum angle.  A right  prism  with  a square  base,  has,  in  like, 
manner,  each,  solid  angle  measured  by  90°  + 90°+90°  — 180° 
= 90°,  which  is  i of  the  maximum  angle.  And  thus  may  be 
found,  that  each  solid  angle  of  a right  prism,  with  an  equilateral. 


triangular  base 

is  } max.  angle 

= 

i .1000. 

square  base 

isi  . . . . 

= 

1 .1000. 

pentagonal  base 

is  .... 

= 

t30  -1000. 

hexagonal 

is  £ • • • • 

= 

Tv  1000. 

heptagonal 

is  .... 

= 

fV  -1000. 

octagonal 

is  f . . . . 

= 

fv  -1000: 

nonagonal 

is  .... 

= 

Vs  1000. 

decagonal 

is  | . . . . 

= 

W iooo. 

undecagonal 

is  .... 

= 

V>_  .1000. 

duodecagonal 

is  x%  ... 

= 

,«  -iooo. 

in  gonal 

is  .... 

— 1000. 
2 m 

Hence  it  may  be  deduced,  that  each  solid  angle  of  a regu- 
lar prism,  with  triangular  base,  is  half  each  solid  angle  of  a 
prism  with  a regular  hexagonal  base.  Each  with  regular 
square  base  — § of  each,  with  regular  octagonal  base, 

pentagonal  = £ decagonal, 

haxagonal 


gonal 


£ duodecagonal, 

m — 4 


m — 2 


m gonal  base. 


Hence  again  we  may  infer,  that  the  sum  of  all  the  solid 
angles  of  any  prism  of  triangular  base,  whether  that  base  be 
regular  or  irregular,  is  half  the  sum  of  the  solid  angles  of  a 
prism  of  quadrangular  base,  regular  or  irregular.  And,  the 
sum  of  the  solid  angles  of  any  prism  of 
tetragonal  base  is  = §•  sum  of  angles  in  prism  of  pentag.  base, 

pentagonal  . • . = £ haxagonal, 

• — i heptagonal, 

• ~ “i (m+!)  gonal. 

2.  Let  us  compare  the  solid  angles  of  the  five  regulai 

bodies.  In  these  bodies,  if  m be  the  number  of  sides  of  each 
face  ; n the  number  of  planes  which  meet  at  each  solid  angle  ; 
\ O = half  the  circumference  or  180°  ; and  a the  plane  angle 

l 

cos  — O 
2n 

made  by  two  adjacent  faces  : then  we  have  sin  ±a  — . 


haxagonal 
:n  gonal 


sin  — O 
2 m 

This 


38 


SPHERICAL  TRIGONOMETRY. 


This  theorem  gives,  for  the  plane  angle  formed  by  every  two 
contiguous  faces  of  the  tetraedron,  70°  31'  42"  ; of  the  rexae- 
dron,  90°  ; of  the  octaedron,  109°  28'  18";  of  the  dodecaedron, 
116°  33'  54'1  ; of  the  icosaedron,  138°  If  23'.  But  in  these 
polyedrae,  the  number  of  faces  meeting  about  each  solid  angle, 
3,  3,  4,  3,  5 respectively.  Consequently  the  several  solid  an- 
gles will  be  determined  by  the  subjoined  proportions  : 

Sol  • An',  le 

360°  : 3-70°31'42"  180°  : : 1000  : 87-73611  Tetraedron. 

360°  : 3-90°  —180°  : : 1000  : 250  ■ Haxaedron. 

360°  : 4-109D28T8'' — 360°  : : 1000  : 216-35185  Octaedron. 

360°  : 3-1 16°33'54'' — 180°  : : 1000  : 471  395  Dodecaedron. 

360°  : 5*138°11'23"— 540°  : : 1000  : 419-30169  Icosaedron. 

3.  The  solid  angles  at  the  vertices  of  cones,  will  be  deter- 

mined by  means  of  the  spheric  segments  cut  otf  at  the  bases' 
of  those  cones  ; that  is,  if  right  cones,  instead  of  having  plane 
bases,  had  bases  formed  of  the  segments  of  equal  spheres, 
whose  centres  were  the  vertices  of  the  cones,  the  surfaces  of 
those  segments  would  be  measures  of  the  solid  angles  at  the 
respective  vertices.  Now,  the  surfaces  of  spheric  segments, 
are  to  the  surface  of  the  hemisphere,  as  their  altitudes,  to  the 
radius  of  the  sphere  ; and  therefore  the  solid  angles  at  the 
vertices  of  right  cones  will  be  to  the  maximum  solid  angle, 
as  the  excess  of  the  slant  side  above  the  axis  of  the  cone,  to 
the  slant  side  of  the  cone.  Thus,  if  we  wish  to  ascertain  the 
solid  angles  at  the  ver.ices  of  the  equilateral  and  the  right- 
angled  cones  , the  axis  of  the  former  is  ^ 3,  of  the  latter, 

2 V the  slant  side  of  each  being  unity.  Hence, 

Angle  at  \ertex. 

1 : 1 — 4 \/  3 : : 1000  : 133  97464,  equilateral  cone, 

1 : 1 — a y/  2 : : 1000  : 292-89322,  right-angled  cone. 

4.  Prom  what  has  been  said,  the  mode  of  determining  the 
solid  angles  at  the  vertices  of  pyramids  will  be  sufficiently  ob- 
vious. If  the  pyramids  be  regular  ones,  if  n be  the  number 
of  faces  meeting  about  the  vertical  angle  in  one,  and  a the 
angle  of  inclination  of  each  two  of  its  plane  faces  ; if  n be  the 
number  of  planes  meeting  about  the  vertex  of  the  other,  and 
a the  angle  of  inclination  of  each  two  of  its  faces  : then  will 
the  vertical  angle  of  the  former,  be  to  the  vertical  angle  of  the 
latter  pyramid,  as  na  — (n — 2)  180°,  to  na  — (n  — 2)  180°. 

If  a cube  be  cut  by  diagonal  planes,  into  6 equal  pyramids 
with  square  bases,  their  vertices  all  meeting  at  the  centre  of 
the  circumscribing  sphere  ; then  each  of  the  solid  angles, 
made  by  the  four  planes  meeting  at  each  vertex,  will  be  i of 
the  maximum  solid  angle  ; and  each  of  the  solid  angles, 
at  the  bases  of  the  pyramids,  will  be  of  the  maximum  solid 

angle 


SPHERICAL  TRIGONOMETRY. 


3B 


angle  Therefore,  each  solid  angle  at  the  base  of  such  pyra- 
mid, is  one-fourth  of  the  solid  angle  at  its  vertex  : and,  if  the 
angle  at  the  vertex  be  bisected,  as  described  below,  either  of 
the  solid  angles  arising  from  the  bisection,  will  be  double,  of 
either  solid  angle  at  the  base.  Hence  also,  and  from  the  first 
subdivision  of  this  scholium,  each  solid  angle  of  a prism,  with 
equilateral  triangular  base,  will  be  half  each  vertical  angle  of 
these  pyramids,  and  double  each  solid  angle  at  their  bases. 

The  angles  made  by  one  plane  with  another,  must  be  ascer- 
tained, either  by  measurement  or  by  computation,  according 
to  circumstances.  But,  the  general  theory  being  thus  explain- 
ed, and  illustrated,  the  further  application  of  it  is  left  to  the 
skill  and  ingenuity  of  geometers  ; the  following  simple  example 
merely,  being  added  here. 

Ex.  Let  the  solid  angle  at  the  vertex  of  a square  pyramid 
be  bisected. 

1st.  Let  a plane  he  drawn  through  the  vertex  and  any  two 
opposite  angles  of  the  base,  that  plane  will  bisect  the  solid  an- 
gle at  the  vertex  ; forming  two  trilateral  angles,  each  equal  t© 
half  the  original  quadrilateral  angle. 

2dly.  Bisect  either  diagonal  of  the  base,  and  draw  any  plane 
to  pass  through  the  point  of  bisection  and  the  vertex  of  the 
pyramid  ; such  plane,  if  it  do  not  coincide  with  the  former, 
will  divide  the  quadrilateral  solid  angle  into  two  equal  quadri- 
lateral solid  angles.  For  this  plane,  produced,  will  bisect  the 
great  circle  diagonal  of  the  spherical  parallelogram  cut  off  by 
the  base  of  the  pyramid  ; and  any  great  circle  bisecting  such 
diagonal  is  known  to  bisect  the  spherical  parallelogram,  or 
square  ; the  plane,  therefore,  bisects  the  solid  angle. 

Cor.  Hence  an  indefinite  number  of  planes  may  be  drawn, 
each  to  bisect  a given  quadrilateral  solid  angle. 

SECTION  II, 

Resolution  of  Sperhical  Triangles. 

The  different  cases  of  spherical  trigonometry,  like  those  in 
plane  trigonometry,  may  be  solved  either  geometrically  or  al- 
gebraically. We  shall  here  adopt  the  analytical  method,  as 
well  on  account  of  its  being  more  compatible  with  brevity, 
as  because  of  its  correspondence  and.  connection  with  the  sub- 
stance 


40. 


SPHERICAL  TRIGONOMETRY, 


stance  of  the  preceding  chapter.*  The  whole  doctrine  may 
be  comprehended  in  the  subsequent  problems  and  theorems 

PROBLEM  I. 


To  Find  Equations,  from  which  may  be  deduced  the  solution  of 
all  the  Cases  of  Spherical  Triangles. 

Let  abc  be  a spherical  triangle  ; ad  the  tangent,  and  gd 
the?1  secant,  of  the  arc  ab  ; ae  the  tangent,  and  ge  the  se- 
cant, of  the  arc  ac  ; let 
the  capital  letters  a,  b, 
c,  denote  the  angles  of 
the  triangle,  and  the 
small  letters  a,  b,  c,  the 
opposite  sides  bc,  ac, 
ab.  Then  the  first  equa- 
tions in  art.  6 PI.  Trig, 
applied  to  the  two  triangles  ade,  gde,  give,  for  the  former, 
de2  = tan2  b -f-  tan2  c — tan  b . tan  c . cos  a ; for  the  latter 

DEa  = sec2  b -f-  sec2  c — sec  b . sec  c . cos  a.  Subtracting 

the  first  of  these  equations  from  the  second,  and  observing 
that  sec2  b—  tan2  b — r2  = 1,  we  shall  have,  after  a little 
. sin  b . sin  c cos  a „ 

reduction,  1 -f cos  a— -=0.  Whence 

cos  b . cos  c cos  b . r os  c 

the  three  following  symmetrical  equations  are  obtained  : 
cos  a = cos  b . cos  c + sin  b . sin  c . cos  a i 

cos  b = cos  a . cos  c -j-  sin  a . sin  c . cos  b > (1. 

cos  c = cos  a . cos  b -j-  sin  a . sin  b . cos  c ) 

THEOREM  VH. 


A 


lu  Every  Spherical  Triangle,  the  Sines  of  the  Angles  are  Pro- 
portional to  the  Sines  of  their  Opposite  sides. 

If,  from  the  first  of  the  equations  marked  1,  the  value 
of  cos  Abe  drawn,  and  substituted  for  it  in  the  equation  sin2 
a = 1 —cos2  a,  we  shall  have 

cos*  a -f-  co«9  b-  cns2  c —2  cos  a cos  b ■ cos  c 


= 1 


sin*  b sir,2  c. 


Reducing  the  terms  of  the  second  side  of  this  equation  to  a 
common  denominator,  multiplying  both  numeratoi  and  deno- 
minator by  sin2  a and  extracting  the  sq.  root  there  will  result 
V(1  —cos2  a — co*2  b —co -j - 2 cos  a.  cos  b.  cos  r.) 
sin  a.  on  b ■ sin  c. 


sin  a = sin  a 


* For  the  geometrical  method,  the  reader  may  consult  Simson’s  or 
Playfair’s  Euclid,  or  Bishop  Horsley’s  Elementary  Treatises  on  Prac- 
tical Mathematics- 


Here 


SPHERICAL  TRIGONOMETRY. 


41 


Here,  if  the  whole  fraction  which  multiplies  sin  a,  be  denoted 
by  k (see  ari.  8 chap,  iii),  we  may  write  sin  a = k . sin  a. 
And,  since  the  fractional  factor,  in  the  above  equation,  con- 
tains terms  in  which  the  sides  a,  b,  c,  are  alike  affected,  we 
have  similar  equations  for  sin  b,  and  sin  c.  That  is  to  say, 
we  have  , 

sin  a = k . sin  a _ . . . sin  b = k . sin  b . . . sin  c = k . sin  c. 

, sm  a sm  b siu  c ., 

Consequently,  - — - = . . . (11.)  which  is  the 

sm  o sm  b sm  c ' ' 

algebraical  expression  of  the  theorem. 

THEOREM  VHI. 

In  Every  Right-Angled  Spherical  Triangle,  the  Cosine  of 
the  Hypothenuse,  is  equal  to  the  Product  of  the  Cosines 
of  the  Sides  Including  the  risht  angle. 

For,  if  a be  measured  by  {Q,  its  cosine  becomes  nothing, 
and  the  first  of  the  equations  i becomes  cos  a = cos  b . cos  c. 
<1  E.  D.  THEOREM  IX. 

In  Every  Right-Angled  Spherical  Triangle,  the  Cosine  of 
either  Oblique  Angle,  is  equal  to  the  Quotient  of  the 
Tangent  of  the  Adjacent  Side  divided  by  the  Tangent  of 
the  Hypothenuse. 

If,  in  the  second  of  the  equations  i,  the  preceding  value  of 
eos  a be  substituted  for  it,  and  for  sin  a its  value  tan  a . cos  a= 
cos  a . cos  b . cos  c ; then  recollecting  that  1 — cos2  c=sin2  c, 
there  will  result,  tan  a . con  c . cos  b = sin  c : whence  it 
follows  that, 

, . tan  c 

tan  a . cos  b = tan  c,  or  cos  b = . 

tan  a 

Thus  also  it  is  found  that  cos  c 

tan  a 

THEOREM  X. 

In  Any  Right-Angled  Spherical  Triangle,  the  Cosine  of  one 
of  the  Sides  about  the  right  angle,  is  equal  to  the  Quotient 
of  the  Cosine  of  the  Opposite  angle  divided  by  the  sine  of 
the  Adjacent  angle. 

From  th.  7,  we  h'aveS'^-^-=s^-^- ; which,  when  a is  a right 

sin  a sin  a ° 

angle,  becomes  simply  sin  b = — . Again,  from  th.  9,  we 

1 sin  a ° 

have  cos  c = — — Hence  by  division, 
tan  a J 

cos  c tan  b sin  a cos  a 

sin  b sir.  b tan  a cos  b 

Now,  th.  8 gives = cos  c.  Therefore  ^-^-=cos  b ; and 

cos  c sin  b 

1-1  cos  B 

*n  like  manner,  = cos  b. 

sin  c 

Yob.  If. 


7 


Q.  E.  D. 

THEOREM 


42 


SPHERICAL  TRIGONOMETRY. 


THEOREM  XI. 

In  Every  Right-Angled  Spherical  Triangle,  the  Tangent  of 
either  of  the  Oblique  Angles,  is  equal  to  the  Quotient  of 
the  Tangent  of  the  Opposite  Side,  divided  by  the  sine  of 
the  Other  Side  about  the  right  angle. 

r,  . sin  b , tan  c 

r or,  since  sin  b — , and  cos  b = , 

sm  a tan  a 

, sin  b sin  b tan  a 

we  nave = . . 

cos  b sin  a tan  c 

Whence,  because  (th.  8)  cos  a = cos  b . cos  c,  and  since 
sin  a — cos  a . tan  a,  we  have 


. sin  b sin  b 

_sin  b 1 

tan  b 

cos  a • tan  c cos  b . cos  c . tan  c 

In  like  manner,  tan  c — Un.i. 

sm  b 

cos  b' cos  c . tan  c 
Q. 

sin  c 
E.  D 

THEOREM  XU. 


In  Every  Right-Angled  Spherical  Triangle,  the  Cosine  of  the 
Hypothenuse,  is  equal  to  the  Quotient  of  the  Cotangent  of 
one  of  the  Oblique  Angles,  divided  by  the  Tangent  of  the 
Other  Angle. 


For,  multiplying  together  the  resulting  equations  of  the 
preceding  theorem,  we  have 

tan  b tan  c 1 

tan  b . tan  c =— — - . — — == . 

sin  b sin  c cos  u . cos  c 

But,  by  th.  8,  cos  b . cos  c = Cos  a. 

1 cot  c 

Therefore  tan  b . tan  c = , or  cos  a = . q.  e.  d 

cos  a tan  b 


THEOREM  XIII. 


In  Every  Right-Angled  Spherical  Tiiangle,  the  Sine  of  the 
Difference  between  the  Hypothenuse  and  Base,  is  equal  to 
the  Continued  Product  of  the  Sine  of  the  Perpendicular. 
Cosine  of  the  Base,  and  Tangent  of  Half  the  Angle  Oppo- 
site to  the  Perpendicular  ; or  equal  to  the  Continued  Pro- 
duct of  the  Tangent  of  the  Perpendicular,  Cosine  of  the 
Hypothenuse,  and  Tangent  of  Half  the  Angle  Opposite  to 
the  Perpendicular*. 


* This  theorem  is  due  to  M.  Prony,  who  published  it  without  de- 
monstration in  the  Connaissance  des  Temps  for  the  year  1808,  and  made 
use  of  it  in  the  construction  of  a chart  of  the  course  of  the  Po. 

Here  ^ 


SPHERICAL  TRIGONOMETRY. 


43 


Here,  retaining  the  same  notation,  since  we  have 

sin  a = and  cos  b = Lria-— ; if  for  the  tangents  there  be 

sin  b tan  a 

substituted  their  values  in  sines  and  cosines,  there  will  arise, 

s:n  b 

sin  c . cos  a = cos  b . cos  c . sin  a = cos  b . cos  c . . 

sin  B 

Then  substituting  for  sin  a,  and  sin  c . cos  a,  their  values  in 
the  known  formula  (equ.  v chap,  iii)  viz. 

in  sin  (a  — c)  = sin  a . cos  c — cos  a . sin  c, 

and  recollecting  that  1 ~c  >s  B = tan  Lb, 

it  will  become,  sin  (a— c)  = sin  b . cos  c . tan^B, 
which  is  the  first  part  of  the  theorem  : and,  if  in  this  result 

we  introduce,  instead  of  cos  c,  its  value  C°S-~  (th.  8),  it  will 

cos  b ' 

be  transformed  into  sin  ( a—c ) =»tan  b . cos  a . tan  ^b  ; which 
is  the  second  part  of  the  theorem.  q.  e.  d. 

Cor.  This  theorem  leads  manifestly  to  an  analogous  one 
with  regard  to  rectilinear  triangles,  which,  if  h,  b,  and  p de- 
note the  hypothenuse,  base,  and  perpendicular,  and  b,  p,  the 
angles  respectively  opposite  to  b,p  ; may  be  expressed  thus  : 

h — b — p ■ tan  ^p h — p = b . tan  ^b. 

These  theorems  may  be  found  uselul  in  reducing  inclined  lines 
to  the  plane  of  the  horizon. 

• PROBLEM  n. 

Given  the  Three  Sides  of  a Spherical  Triangle;  it  is  re- 
quired to  find  Expressions  for  the  Determination  of  the 
Angles. 

Retaining  the  notation  of  prob.  1 , in  all  its  generality,  we 
soon  deduce  from  the  equations  marked  i in  that  problem,  the 
following  ; viz. 

cos  a — cos  b . cos 

cos  a = 

sin  b . sin  c 

cos  b — cos  a . cos 

COS  B ==  — — : 

sin  a . sin  c 
cos  c— cos  a . cos 

COS  C = 

sin  a - sin  b 

As  these  equations,  however,  are  not  well  suited  for  loga- 
rithmic computation  ; they  must  be  so  transformed,  that  their 
second  members  will  resolve  into  factors.  In  order  to  this, 
substitute  in  the  known  equation  1 — cos  a = 2 sin3  4-a,  the 
preceding  value  of  cos  a,  and  there  will  result 

2 sin3  J-a  = c)~ cos  « 

sin  b ■ sin  c 

But,  because  cos  b'  — cos  a'  = 2 sin  | (a'  + b')  . sin  ^(a'— b') 
(art.  25  ch.  iii),  and  consequently,  ’ cos 


44 


SPHERICAL  TRIGONOMETRY. 


cos  ( b — c ) — cos  a = 2 sin  — - . sin  — - 

2 2 

we  have,  obviously, 

sin2  i-\  = i(a+6~ 0 • sin  j(o+  c -6) 

sin  6 • sm  c 

Whence,  makings  = a + 6 + c,  there  results 
sin  1a  = ^ ~6)  • s,n  (*s-0. 


r.  , . sin  ( 4 

So  also,  sin  iB  = ^/  - v 


sin  b . sin  c 

fl)  . sin  (£s—  c) 


(111.) 


And,  sin  ic  = ^/ 


sin  a • sin  c 

sin  (Js  — a)  . sin  (£s  — 6) 
sin  a . sin  b 

The  expressions  for  the  tangents  of  the  half  angles,  might  have 
been  deduced  with  equal  facility  ; and  we  should  have  obtain- 
ed, for  example, 

tan  y sin(^-sin(*sTf).  (Hi  ) 

^ sin  • sin  $ (s  — a) 

Thus  again,  the  expressions  for  the  cosine  and  cotangent  of 
half  one  of  the  angles,  are 

, sin  As  . sin  Afs—  o') 

COS  J-A  = ./  2 

' sin  b ■ sin  c 

sin  . sin  i (s— a) 

cot  4a  = */ — 7\ r- 

2 v sin  — b)  . sin  (js— 

The  three  latter  flowing  naturally  from  the  former,  by  means 

sin  , cos 

, cot  = — 

cos  sin 

Cor.  1.  When  two  of  the  sides,  as  b and  c,  become  equal, 
then  the  expression  for  sin  \a  becomes 
sin(A- — b ) sin  ia 

Sin  iA  — — — , - = r • 

s n o sin  b 

When  all  the  three  sides  are  equal,  or  a = b = cs 

sin  i<j 


of  the  values  tan  = — , cot — . (art.  4 ch.  iii.) 

rnc  Sin  v ' 


Cor.  2. 
then  sin  4a 


Cor.  3 
W* 


In  this  case*  if  a — b = c — S0°  ; then  sin  4a= 
= 4^/2  = sin  45°  : and  a = e = c = 90°. 

Cor.  4.  If  a = b = c ==  60°  : then  sin  Aa  = —^r=±x/3= 

2V  J 

sin  35°15'51"  : and  a = b = c = 70°31'42",  the  same  as  the 
angle  between  two  contiguous  planes  of  a tetraedron. 

Cor.  5.  If  a = b = c were  assumed  = 1 20°  : then  sin  4a  = 

sm  6j  — 2 v'ji  _ j . an(j  A __  B _ c = i go0  : which  shows 


sin  12G°  4v/3 

that  no  such  triangle  can  be  constructed  (conformably  to 
th  2)  ; but  that  the  three  sides  would,  in  such  case,  form  three 
continued  arcs  completing  a great  circle  of  the  sphere. 


PROBLEM 


SPHERICAL  TRIGONOMETRY. 


4S 


PROBLEM  III. 


Given  the  Three  Angles  of  a Spherical  Triangle,  to  find 
Expressions  for  the  Sides. 

If  from  the  first  and  third  of  the  equations  marked  I 
(prob.  I),  cos  c be  exterminated,  there  will  result, 

cos  a . sin  c -{-  cos  c . sin  a . cos  b = cos  a . sin  b. 

But,  it  follows  from  th.  7,  that  sin  c = siu  a . sin  c_.  gubstitut- 

sin  a 

ing  for  sin  c this  value  of  it,  and  for  °°S  \ their  equiva- 

sm  a sin  a 

lents  cot  a,  cot  a,  we  shall  have, 

cot  a . sin  c -f-  cos  c . cos  b = cot  a . sin  b. 

. . , cos  a sin  b sin  B 

Mow,  cot  a . sin  b = , sin  o — cos  a . = cos  a . , 

stn  a sin  a sin  A 

(th.  7).  So  that  the  preceding  equation  at  length  becomes, 
cos  a sin  c = cos  a . sin  b — sin  a . cos  c . cos  b. 

In  like  manner,  we  have, 

cos  b . sin  c = cos  b . sin  a — sin  b . cos  c . cos  a. 
Exterminating  cos-  b from  these,  there  results 


cos  a = cos  a . sin  b sin  c — cos  b . cos  c.  i 

So  like-  £ cos  e = cos  b . sin  a sin  c — cos  a . cos  c.  > (IV.) 

wise  $ cos  c = cos  c . sin  a sin  b — cos  a . cos  b.  ) 

This  system  of  equations  is  manifestly  analogous  to  equa- 

tion i ; and  if  they  be  reduced  in  the  manner  adopted  in  the 
last  problem,  they  will  give 


sin  \a  — ^/ 

sin  ±b  = y/ 

sin  \c  = 
i expression 
tan  |a  = y/ 


sin  b 

. sin 

c 

COS 

i(A  J-  B-l-Ci 

•1(a+c- 

-B) 

sill  A 

. sin 

c 

cos 

t 

^(a+  b-i-c) 

. cos 

«A  + B 

-c) 

sin  a 

sin 

B 

for  the  tangent  of 

half 

’ a side 

is 

r COS 

i(A  + B + c) 

. COS 

Kb+o- 

-A) 

•J 


(V). 


COS  i(A-pC—  b)  . COS  £(A-f  B-  c)' 

The  values  of  the  cosines  and  cotangents  are  omitted,  to 
save  room  ; but  are  easily  deduced  by  the  student. 

Cor.  1.  When  two  of  the  angles,  as  b and  c,  become  equal, 

when  the  value  of  cos  \a  becomes  cos  ±a  — c°s- 

sin  b 

Cor.  2.  When  a = b = c ; then  cos  la  = C-b  2-A~. 

s n a 

Cor.  3.  When  a = b = c = 90°,  then  a — b = c = gO". 


sin  60° 


Cor.  4,  If  a = b = c = 60°  : then  cos 

’ 2 sin  60 

So  that  a — b = c = 0.  Consequently  no  such  triangle  can 
be  constructed  : conformably  to  th.  3.  Cor. 


46 


SPHERICAL  TRIGONOMETRY. 


/*nc  60®  A 

Cor.  5.  If  a=b=c=  120°:  then  cos  4a  = — = — — =: 

sin  120°  iv/3 

i 3 = cos  64°  44'  9".  Hence  a = b = c = 109°  2b'  18". 

Sc4o/  If  in  the  preceding  values  of  sin  \a,  sin  \b,  &c.  the 
quantities  under  the  radical  were  negative  in  reality,  as  they 
are  in  appearance,  it  would  obviously  be  impossible  to  deter- 
mine the  value  of  sin  \a , &c.  But  this  value  is  in  fact  always 
real.  For,  in  general,  sin  (x  — i Q)  = — cos  x : therefore 

sin—— { O)"  — cos  i(a  + b + c)  ; a quantity  which 


is  always  positive,  because,  as  a + b + c is  necessarily  com- 
prised between  40  and  fO>  we  have  |(a  + b + c)  — {O 
greater  than  nothing,  and  less  than  40  Further,  any  one 
side  of  a spherical  triangle  being  smaller  than  the  sum  of  the 
other  two,  we  have,  by  the  property  of  the  polar  triangle 
(theorem  4),  40  — A less  than  40  — b + 40  — c ; whence 
4 (b  -f-  c — a)  is  less  than  ]0  ; and  of  course  its  cosine  is 
positive. 


PROBLEM  IV. 


Given  Two  Sides  of  a Spherical  Triangle  and  the  Included 
Angle  to  obtain  Expressions  for  the  Other  Angles. 

1.  In  the  investigation  of  the  last  problem,  we  had 
cos  a . sin  c = cos  a . sin  b — cos  c . sin  a . cos  b : 
and  by  a simple  permutation  of  letters,  we  have 

cos  b . sin  c = cos  6 . sin  a — cos  c . sin  b . cos  a : 
adding  together  these  two  equations,  and  reducing,  we  have 
sin  e (cos  a + cos  b)  = (1  — cos  c)  sin  {a  4-  b). 

Now  we  have  from  theor.  7, 

sin  a sin  c an(jsin  b sin  c 

sin'  A sin  c’  sin  B sin  c 

Freeing  these  equations  from  their  denominators,  and  respect- 
ively adding  and  subtracting  them,  there  results 

sin  c (sin  a 4*  fin  b)  = sin  c (sin  a 4-  sin  b ) 
and  sin  c (sin  a — sin  b)  = sin  c (sin  a — sin  b). 
Dividing  each  of  these  two  equations  by  the  preceding,  there 
will  be  obtained 


sin  a 4 sin  B 
cos  a -f  cos  B 
sin  a — sin  b 


sin  c sin  a 4.  sin  b 

1 — cos  c sin  (</  -f  b)  ’ 

sin  c sin  a — sin  b 
1—  cos  c sin  (a  -p  b) 


cos  a 4"  cos  B 

Comparing  these  with  the  equations  in  arts.  25,  26,  27,  ch.  iii, 
there  will  at  length  result 

tan  i(a  4- b)  = cot  4c.  

» ' - cos  i(o-H) 

tan  4(a  — b)  = cot  £c.  ■ — 2 v — 


(VI.) 


sin  2tfl4~6) 


Cor. 


SPHERICAL  TRIGONOMETRY. 


47 


Cor.  When  a = b,  the  first  of  the  above  equations  be- 
comes tan  a = tan  b = cot  |c  sec  a. 

And  in  this  case  it  will  be,  as  rad  : sin  : : sin  a or  sin  b : 
sin  \c. 

And,  as  rad  : cos  a or  cos  b : : tan  a or  tan  b : tan  ±c. 

2.  The  preceding  values  of  tan  t(a  + b),  tan  i(a — b)  are 
very  well  fitted  for  logarithmic  computation  : it  may,  notwith- 
standing, be  proper  to  investigate  a theorem  which  will  at  once 
lead  to  one  of  the  angles  by  means  of  a subsidiary  angle.  In 
order  to  this,  we  deduce  immediately  from  the  second  equation 
in  the  investigation  of  prob.  3, 

cot  a . sin  b , , 

cot  A = cot  c . cos  b. 

sin  c 

Then,  choosing  the  subsidiary  angle  <p  so  that 
tan  <p  = tan  a . cos  c, 

that  is,  finding  the  augle  <p,  whose  tangent  is  equal  to  the  pro- 
duct tan  a . cos  c,  which  is  equivalent  to  dividing  the  original 
triangle  into  two  right-angled  triangles,  the  preceding  equation 
will  become 

cot  A=cot  c(cotp.sin  b — cos 6)=*^—  (cos (f> . sin b — sin^  . cos  b). 

And  this,  since  sin  (b — <p ) = cos  <p . sin  b — sin  <p  . cos  b becomes 

cot  c . . 

cot  a = — . sin  Cb — <p). 

sin  ? v 

Which  is  a very  simple  and  convenient  expression. 

PROBLEM  V. 


'Given  Two  Angles  of  a Spherical  Triangle,  and  the  Side 
Comprehended  between  them  ; to  find  Expressions  for 
the  Other  Two  Sides. 


1.  Here,  a similar  analysis  to  that  employed  in  the  preced- 
ing problem,  being  pursued  with  respect  to  the  equations  iv, 
in  prob.  3,  will  produce  the  following  formulae 

sin  a + sin  b sin  c sin  a + sin  B 

cos  a + cos  b l + cos  c'sin  (a  -j-  b)  ’ 
sin  a — sin  6 sin  c sin  A — sin  B 


cos  a -f  cos  b 1 -f-  cos  c sin  (a  -f-  b) 
Whence,  as  in  prob.  4,  we  obtain 

cos  £(a—  b) 


tan  i(a+6)  = tan^c. 
tan  |(a  — b)  = tan^c. 


COS  ^(A-f-B 
sin  ^(a 


sin  i(A-j-  b 


1) 

) > 

■) ) 


(VII*) 


2.  If 


* The  formulae  marked  vi,  and  vn,  converted  into  analogies,  by 
making  the  denominator  of  the  second  member  the  fi  st  term,  the 
othe"  two  factors  the  second  and  third  terms,  and  the  first  member  of 
. the  equation,  the  fourth  term  of  the  proportion,  as 


cos 


48 


SPHERICAL  TRIGONOMETRY. 


2.  If  it  be  wished  to  obtain  a side  at  once,  by  means  of  a 
subsidiary  angle  ; then,  find  <p  so  that  C°  A=  tan  tp  ; then  will 


, cot  C , .N 

cot  a= . cos  (b  —<P). 

cos  f ' ' 


PROBLEM  VI. 


Given  Two  Sides  of  a Spherical  Triangle,  and  an  Angle  Op- 
posite to  one  of  them  ; to  find  the  Other  Opposite  Angle. 
Suppose  the  sides  given  are  a,  b,  and  the  given  angle  b : 

then  from  theor.  7,  we  have  sin  a— **■——- -b"—  • or,  sin  a,  a 

sin  b 

fourth  proportional  to  sin  b,  sin  b,  and  sin  a. 


PROBLEM  VII. 

Given  Two  Angles  of  a Spherical  Triangle,  and  a Side 
Opposite  to  one  of  them  ; to  find  the  Side  Opposite  to  the 
other. 

Suppose  the  given  angles  are  a,  and  b,  and  b the  givea 
side  : then  th.  7,  gives  sin  a __s'n  b • s.n  a _ ^ sjn  Q)  a fourth 


proportional  to  sin  b sin  b,  and  sin  a. 

Scholium- 

In  problems  2 and  3,  if  the  circumstances  of  the  question 
leave  any  doubt,  whether  the  arcs  or  the  angles  sought,  are 
greater  or  less  than  a quadrant,  or  than  a right  angle,  the 
difficulty  will  be  entirely  removed  by  means  of  the  table  of 
mutations  cf  signs  of  trigonometrical  quantities,  in  different 
quadrants,  marked  vii  in  chap  3 In  the  6th  and  7th  pro- 
blems, the  question  proposed  will  often  be  susceptible  of  two 
solutions  : by  means  of  the  subjoined  table  the  student  may 
always  tell  when  this  will  or  will  not  be  the  case 

1.  With  the  data  a,  b,  and  b,  there  can  only  be  one  solution 
when  b = i O (a  right  angle), 
or,  when  B<\0-...a<±0....b>a, 

B<jO  • - - - a >^0  ....n  > j O — a. 


cos  4 (a  ~\-b)  : cos  ^(c — b)  : : cot  jc  : tan  -§(  v b), 
sin  4(a  -J- b)  : sin  4(«  — i)  : : cot  4c  : tan  ^(a  — b)>  &c.  &c. 
are  called  the  Analogies  of  Napier , being  invented  b , that  celebrated 
geometer.  He  likewise  invented  other  rules  for  spherical  trigonome- 
try, known  by  the  name  of  Napier’s  Rules  for  the  circular  parts  ; but 
these,  notwithstanding  their  ingenuity,  are  net  inserted  here  ; be- 
cause they  are  too  artificial  to  be  applied  by  a young  computist,  to 
every  case  that  may  occur,  without  considerable  danger  of  misappre- 
hension and  error. 

These  objections  to  Napier’s  rules  do  not  appear  to  me  to  be 
well  founded.  Adraiio 

The 


SPHERICAL  TRIGONOMETRY. 


4? 


i be  triangle  is  susceptible  of  two  forms  and  solutio  n 
when  b<  JO  • • • • «<  JO  . ...  b < a, 

b<±0  ....  a>  ±0  ....  b < ± O—a, 

b>JO «<  ?0  ....  6 > 

b>JO  • • • • a>  JO  . . • . 6 > a, 

b < ar  > J O • • • • a~  JO- 

2.  With  the  data  a,  b,  and  b,  the  triangle  can  exist,  but  in. 
one  form, 

when  6=JO  (one  quadrant), 

6>JO  . ...  a > JO  ....  b < A, 

6>J O a<  JO b <1  O—a, 

£<JO  ....  A > JO  ■ • • • B >JO-A, 

6<JO  . • . . A < jo  . • • • B >A. 

It  is  susceptible  of  two  forms, 
when  6>JO  ....a>JO  ....b  >a, 

b> JO a<  JO  . . . . b > J O—a, 

b < JO  ..*.A>JO  « • • . B < J O-A, 

6<JO  ....  A<  JO  ....  B < A, 
b < or>  JO  . ...  a = J O • 

It  may  here  be  observed,  that  all  the  analogies  and  formulas, 
of  spherical  trigonometry,  in  which  cosines  or  cotangents  are 
not  concerned,  may  be  applied  to  plane  trigonometry  ; taking 
care  to  use  only  a side  instead  of  the  sine  or  the  tangent  of  a 
side  ; or  the  sum  or  difference  of  the  sides  instead  of  the  sine 
or  tangent  of  such  sum  or  difference.  The  reason  of  this  is 
obvious  : for  analogies  or  theorems  raised:  not  only  from  the 
consideration  of  a triangular  figure,  but  the  curvature  of  the 
sides,  also,  are  of  consequence  more  general  ; and  therefore, 
though  the  curvature  should  be  deemed  evanescent,  by  reason 
of  a diminution  of  the  surface,  yet  what  depends  on  the  tri- 
angle alone  will  remain  notwithstanding. 

We  have  now  deduced  all  the  rules  that  are  essential  in 
the  operations  of  spherical  trigonometry  ; and  explained  un- 
der what  limitations  ambiguities  may  exist.  That  the  student, 
however,  may  want  nothing  further  to  direct  his  practice  in 
this  branch  of  science,  we  shall  add  three  tables,  in  which  the 
several  formulae,  already  given,  are  respectively  applied  to  the 
solution  of  all  the  cases  of  right  and  oblique-angled  spherical 
triangles,  that  can  possibly  occur. 


Vol.  II. 


ft 


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For  the  Solution  of  all  the  cases  of  Oblique-Jingled  Spherical  Triangles,  by  the  Analogies  of  JVapier. 


54 


SPHERICAL  TRIGONOMETRY 


| Third  side.  < By  the  common  analogy. 


SPHERICAL  TRIGONOMETRY 


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56 


SPHERICAL  TRIGONOMETRY. 


Questions  for  Exercise  in  Spherical  Trigonometry. 

Ex.  1.  In  the  right-angled  spherical  triangle  bac,  right- 
angled  at  a,  the  hypothenuse  a = 78°20'.  and  one  leg  c = 
76°52',  are  given  : to  find  the  angles  b,  and  c,  and  the  other 
leg  b. 

Here,  by  table  i case  1,  sin  c = ; 

sin  a 

tan  c , cos  a 

cos  b = ; . . . . cos  b — . 

tan  a cos  c 

Or,  log  sin  c = log  sin  c — log  sin  a -f-  10 
log  cos  b = log  tan  c — log  tan  a -f-  10. 
log  cos  b = log  cos  a — log  cos  c+  10. 

Hence,  10  + log  sin  c — >0  -j-  log  sin  76°52'  = 19-9884894 
log  sin  a = log  sin  78°20'  = 9-9909338 


Remains,  log  sin  c = log  sin  83°56'  = 9-9975556 


Here  c is  acute,  because  the  given  leg  is  less  than  90°. . 
Again,  10  -f-  log  tan  c — 10  + log  tan  76°62'=  20-6320468 
log  tan  a = log  tan  78°20'  = 10-6851149 


Remains,  log  cos  b = log  cos  27°45'  =.  9-9469319 


b is  here  acute,  because  a and  c are  of  like  affection. 

Lastly,  10  + log  cos  a = 10  -f  log  cos  78°20'  = 19  3058189 
log  cos  c = log  cos  76°52'  = 9-3564426 


Remains,  log  cos  b = log  cos  27°8'  = 9-9493763 


where  b is  less  than  90°,  because  a and  c both  are  so. 

Ex.  2.  In  a right-angled  spherical  triangle,  denoted  as 
above,  are  given  a — 78°20',  b = 27°45'  ; to  find  the  other 
sides  and  angle. 

Ans.  b = 27°8',  c = 76°52',  c = 83°56'. 

Ex.  3.  In  a spherical  triangle,  with  a a right  angle,  given 
b = 1 17°34',  c = 31°51'  ; to  find  the  other  parts. 

Ans.  a = 1 13°55',  c 28°51',  b = 104°  8'. 

Ex.  4.  Given  b — 27°6',  c = 76°52’  ; to  find  the  other 
parts.  Ans.  a — 78°20  b = 27°45',  c = 83°56'. 

Ex.  5.  Given  b = 42°  12'  b ==  48°  ; to  find  the  other  parts. 

Ans.  a — t’4°-i0'i,  or  its  supplement, 
c = 54° 4-1 , or  its  supplement, 
c = 64°35  , or  its  supplement. 

Ex  6.  Given  b = 48°,  c = 64°35'  ; required  the  other 
parts  ? Ans.  b = 42°  12',  c = 54°44‘,  a = 64*40'i. 

Ex. 


SPHERICAL  TRIGONOMETRY. 


57 


Ex.  7.  In  the  quadrantal  triangle  abc,  given  the  quad- 
rantal  side  a = 90°,  an  adjacent  angle  c,=  42°  12'  and  the 
opposite  angle  a = 64°  40'  ; required  the  other  parts  of  the 
triangle  ? 

Ex.  8.  In  an  oblique-angled  spherical  triangle  are  given 
the  three  side  viz.  a = 56°  40',  b = 83°  13',  c = 114°  30'  : 
to  find  the  angles. 

Here,  by  the  fifth  case  of  the  table  2,  we  have 

sin  (%s  — 6)  . sin  (As—  c) 

sin  4a  = -2—~: — : — — ■ ■ 

* sin  b ■ sin  c 

Or,  log.  sin  4 A=log  sin  (-| s—b)  + log  sin  (|s — c)  -j-  ar.  comp, 
log.  sin  b -J-  ar.  comp,  log  sin  c : where  s = a -f-  b -}-  c. 
log  sin  (|s  — b)  = log  sin  43°  58'  \ = 9-8415749 
log  sin  (is  — c)  = l°g  sin  12°  41'  a = 9-3418385 
A.  c.  log  sin  b — a . c . log  sin  83°  13'  = 0 0030508 

a.  c.  log  sin  c = a . c . log  sin  1 14°  30'  = 0-0409771 

Sum  of  the  four  logs 19-2274413 

Half  sum  = log  sin  4 a log  sin  24°  15'  \ =9*6137206 
Consequently  the  angle  a is  48°  31' 

Then,  by  the  common  analogy, 

As,  sin  a . . . sin  56°40'  ...  log  = 9*9219401 
To,  sin  a . . . sin  48°31'  ...  log  = 9-8745679 
So  is,  sin  b . . . sin  83°  13'  ...  log  = 9*9969492 

To,  sin  e . . . sin  62°56'  ...  log  = 9-9495770 

And  so  is,  sin  c . . . sin  1 14°30' ...  log  = 9-9590229 
To, sine  . . . sin  125°  19' . . . log  = 9-9116507 
So  that  the  remaining  angles  are,  b = 62°56',  and  c = 125?19'. 


2dly.  By  way  of  comparison  of  methods,  let  us  find  the 
angle  a,  by  the  analogies  of  Napier,  according  to  case  5 table 
3.  In  order  to  which,  suppose  a perpendicular  demitted  from 
the  angle  c on  the  opposite  side  c.  Then  shall  we  have  tan  \ 

dm.  seg.  of  c — ^ — - — — . 

° tan  $c 

This  in  logarithms,  is 

log  tan  i(4  + a)  = log  tan  69°56'  £ = 10-4375601 
log  tan  i (b  — a)  = log  tan  13^16'  ^ — 9-3727819 


Their  sum  = 19-8103420 
Subtract  log  tan  J-c  = log  tan  57°  15'  = 10-1916394 
Rem.  log  cos  dif.  seg  = log  cos  22°  34'  = 9-6187026 
Hence,  the  segments  of  the  base  are  79°  49'  and  34Q  41'. 


Vox..  II. 


9 


Therefore, 


58 


SPHERICAL  TRIGONOMETRY. 


Therefore,  since  cos  a = tan  79°  49‘  X cot  b i 

To  log  tan  adja.  seg.  = log  tan  79°  49'  = 10-7456257 

Add  log  tan  side  b = log  tan  83°  13'  = 9-0753563 

The  sum  rejecting  10  from  the  index  > _ 9.8209820 

log  cos  a = log  cos  48°  32'  $ 

The  other  two  angles  may  be  found  as  before.  The  prefer 
ence  is,  in  this  case,  manifestly  due  to  the  former  method. 

Ex.  9.  In  an  oblique-angled  spherical  triangle,  are  given 
two  sides  equal  to  114°  40’  and  56°  30‘  respectively,  and  the 
angle  opposite  the  former  equal  to  125°  20'  to  find  the  other 
parts.  Ans.  Angles  48°  30',  and  62°  55'  ; side,  83°  12'. 

Ex  10.  Given,  in  a spherical  triangle,  two  angles,  equal 
to  48°  30',  and  125°  20',  and  the  side  opposite  the  latter,  to 
find  the  other  parts. 

Ans.  Side  opposite  first  angle,  56°  40'  ; other  side,  83°  12' 
third  angle  62°  54'. 

Ex.  11  Given  two  sides,  equal  114°  30',  and  56°  40' ; and 
their  included  angle  62°  54'  : to  find  the  rest. 

Ex.  12.  Given  two  angles,  125°20'  and  48°30',  and  the  side 
comprehended  between  them  83°  12'  : to  find  the  other  parts. 

Ex.  13.  In  a spherical  triangle,  the  angles  are  48°31',  62°56', 
and  125°20'  : required  the  sides  ? 

Ex.  14.  Given  two  angles,  50°  12',  and  58°  8'  ; and  aside 
opposite  the  former,  62°  42'  ; to  find  the  other  parts. 

Ana.  The  third  angle  is  either  130°56'  or  156°  14'. 
Side  betw  giv.  angles,  either  119°4'  or  152°  14'. 
Side  opp.  58°8',  either  79°12'  or  100°48'. 

Ex.  15.  The  excess  of  the  three  angles  of  a triangle, 
measured  on  the  earth’s  surface,  above  two  right  angles,  is 
1 second  ; what  is  its  area,  taking  the  earth’s  diameter  at 
7957|  miles  ? 

Ans.  76-75299,  or  nearly  76f  square  miles. 

Ex.  16.  Determine  the  solid  angles  of  a regular  pyramid, 
with  hexagonal  base,  the  altitude  of  the  pyramid  being  te 
each  side  of  the  base  as  2 to  1. 

Ans.  Plane  angle  between  each  two  lateral  faces  126°52'H"^. 

between  the  base  and  each  face  66Q35'12  i. 
Solid  angle  at  the  vertex  114-49768  ) The  max.  angle 
Each  ditto  at  the  base  222-34298  ) being  1000. 


OX  GEODESIC 


I 59  ] 


ON  GEODESIC  OPERATIONS,  AND  THE  FIGURE  OF  THE 
EARTH. 


SECTION  I. 

General  Account  of  this  kind  of  Surveying. 

Art.  1.  In  the  treatise  on  Land  Surveying  in  the  first 
volume  of  this  Course  of  Mathematics,  the  directions  were 
restricted  to  the  necessary  operations  for  surveying  fields, 
farms,  lordships,  or  at  most  counties  ; these  being  the  only 
operations  in  which  the  generality  of  persons,  who  practise 
this  kind  of  measurement,  are  likely  to  be  engaged  : but  there 
are  especial  occasions  when  it  is  requisite  to  apply  the  princi- 
ples of  plane  and  spherical  geometry,  and  the  practices  of  sur- 
veying, to  much  more  extensive  portions  of  the  earth’s  sur- 
face ; and  when  of  course  much  care  and  judgment  are  called 
into  exercise,  both  with  regard  to  the  direction  of  the  practical 
operations,  and  the  management  of  the  computations.  The 
extensive  processes  which  we  are  now  about  to  consider,  and 
which  are  characterised  by  the  terms  Geodesic  Operations  and 
Trigonometrical  Surveying , are  usually  undertaken  for  the  ac- 
complishment of  one  of  these  three  objects.  1.  The  finding 
the  difference  of  longitude,  between  two  moderately  distant 
and  noted  meridians  ; as  the  meridians  of  the  observatories  at 
Greenwich  and  Oxford,  or  of  those  at  Greenwich  and  Paris. 
2.  The  accurate  determination  of  the  geographical  positions 
of  the  principal  places,  whether  on  the  coast  or  inland,  in  an 
island  or  kingdom  ; with  a view  to  give  greater  accuracy  to 
maps,  and  to  accommodate  the  navigator  with  the  actual  posi- 
tion, as  to  latitude  and  longitude,  of  the  principal  promonto- 
ries, havens,  and  ports.  These  have,  till  lately,  been  deside- 
rata, even  in  this  country  : the  position  of  some  important 
points,  as  the  Lizard,  not  being  known  within  seven  minutes 
of  a degree ; and,  until  the  publication  of  the  board  of  Ord- 
nance maps,  the  best  country  maps  being  so  erroneous,  as  in 
some  cases  to  exhibit  blunders  of  three  miles  in  distances  of 
less  than  twenty. 


3.  Th<^ 


60 


TRIGONOMETRICAL  SURVEYING. 


3.  The  measurement  of  a degree  in  various  situations  , and 
thence  the  determination  of  the  figure  and  magnitude  of  the 
earth. 

When  objects  so  important  as  these  are  to  be  attained,  it  is 
manifest  that,  in  order  to  ensure  the  desirable  degree  of  cor- 
rectness in  the  results,  the  instruments  employed,  the  opera- 
tions performed,  and  the  computations  required,  must  each 
have  the  greatest  possible  degree  of  accuracy.  Of  these,  the 
first  depend  on  the  artist  ; the  second  on  the  surveyor  or  en- 
gineer, who  conducts  them  : and  the  latter  on  the  theorist  and 
calculator  : they  are  these  last  which  will  chiefly  engage  our 
attention  in  the  present  chapter. 

2.  In  the  determination  of  distances  of  many  miles,  whether 
for  the  survey  of  a kingdom,  or  for  the  measurement  of  a de- 
gree, the  whole  line  intervening  between  two  extreme  points 
is  not  absolutely  measured ; for  this,  on  account  of  the  inequa- 
lities of  the  earth’s  surface,  would  be  always  very  difficult, 
and  often  impossible.  But,  a lice  of  a few  miles  in  length  is 
very  carefully  measured  on  some  plane,  heath,  or  marsh,  which 
is  so  nearly  level  as  to  facilitate  the  measurement  of  an  actual- 
ly horizontal  line  ; and  this  line  being  assumed  as  the  base  of 
the  operations,  a variety  of  hills  and  elevated  spots  are  select- 
ed gt  which  signals  can  be  placed,  suitably  distant  and  visible 
one  from  another  : the  straight  lines  joining  these  points  con- 
stitute a double  series  of  triangles,  of  which  the  assumed  base 
forms  the  first  side  ; the  angles  of  these,  that  is  the  angles 
made  at  each  station  or  signal  staff,  by  two  other  signal  staffs 
are  carefully  measured  by  a theodolite,  which  is  carried  suc- 
cessively from  one  station  to  another.  In  such  a series  of  tri- 
angles, care  being  always  taken  that  one  side  is  common  to 
two  of  them,  all  tbe  angles  are  known  from  the  observations  at 
the  several  stations,'  and  a side  of  one  of  them  being  given, 
namely  that  of  the  base  measured,  the  side  of  all  the  rest,  as 
well  as  the  distance  from  the  first  angle  of  the  first  triangle  to 
any  part  of  the  last  triangle,  may  be  found  by  the  rules  of  trigo- 
nometry. And  so  again,  the  bearing  of  any  one  of  the  sides, 
with  respect  to  the  meridan,  being  determined  byr  observation, 
the  bearings  of  any  of  the  rest,  with  respect  to  the  same  me- 
ridian, will  be  known  by  computation.  In  these  operations,  it 
is  always  adviseable,  when  circumstances  will  admit  of  it,  to 
measure  another  base  (called  a base  of  verification)  at  or  near 
the  ulterior  extremity  of  the  series  : for  tbe  length  of  this  base. 
computed  as  one  of  the  sides  of  the  chain  of  triangles,  com- 
pared with  its  length  determined  by  actual  admeasurement , 
will  be  a test  of  the  accuracy  of  all  the  operations  made  in  tbe 
series  between  the  two  bases. 


3.  Now 


GEODESIC  OPERATIONS. 


61 


3.  Now,  in  every  series  ef  triangles,  where 
each  angle  is  to  be  ascertained  with  the  same  in- 
strument, they  should,  as  nearly  as  circumstances 
will  permit,  be  equilateral.  For,  if  it  were  pos- 
sible to  choose  the  stations  in  such  manner,  that 
each  angle  should  be  exactly  60  degrees  ; then, 
the  half  number  of  triangles  in  the  series,  multi- 
plied into  the  length  of  one  side  of  either  trian- 
gle would,  as  in  the  annexed  figure,  give  at  once 
the  total  distance  ; and  then  also,  not  only  the 
sides  of  the  scale  or  ladder,  constituted  by  this  se- 
ries of  triangles,  would  be  perfectly  parallel,  but 
the  diagonal  steps,  marking  the  progress  from 
one  extremity  to  the  other,  would  be  alternately 
parallel  throughout  the  whole  length.  Here  too, 
the  first,  side  might  be  found  by  a base  crossing  it  perpendicu- 
larly of  about  half  its  length,  as  at  h ; and  the  last  side  veri- 
fied by  another  such  base,  r,  at  the  opposite  extremity.  If  the 
respective  sides  of  the  series  of  triangles  were  12  or  18  miles, 
these  bases  might  advantageously  be  between  6 and  7,  or  be- 
tween 9 and  10  miles  respectively  ; according  to  circumstances. 
It  may  also  be  remarked,  (and  the  reason  of  it  will  be  seen  in 
the  next  section)  that  whenever  only  two  angles  of  a triangle 
can  be  actually  observed,  each  of  them  should  be  as  nearly 
as  possible  45°.  or  the  sum  of  them  about  90°  ; for  the  less 
the  third  or  computed  angle  differs  from  90°,  the  less  proba- 
bility there  will  be  of  any  considerable  error.  See  prob.  1 
sect.  2,  of  this  chapter. 

4.  The  student  may  obtain  a general  notion  of  the  method 
employed  ia  measuring  an  arc  of  the  meridian,  from  the  fol- 
lowing brief  sketch  and  introductory  illustrations. 

The  earth,  it  is  well  known,  is  nearly  spherical.  It  majr  be 
either  an  ellipsoid  of  revolution,  that  is,  a body  formed  by 
the  rotation  of  an  ellipse,  the  ratio  of  whose  axes  is  nearly 
that  of  equality,  on  one  of  those  axes  ; or  it  may  approach 
nearly  to  the  form  of  such  an  ellipsoid  or  spheroid,  while  its 
deviations  from  that  form,  though  small  relatively , may  still 
be  sufficiently  great  in  themselves  to  prevent  its  being  called 
a spheroid  with  much  more  propriety  than  it  is  called  a sphere. 
One  of  the  methods  made  use  of  to  determine  this  point,  is 
by  means  of  extensive  Geodesic  operations. 

The  earth  however,  be  its  exact  form  what  it  may,  is  a 
planet,  which  not  only  revolves  in  an  orbit,  but  turns  upon 
an  axis.  Now,  if  we  conceive  a plane  to  pass  through  the 
axis  of  rotation  of  the  earth,  and  through  the  zenith  of  any 
place  on  its  surface,  this  plane,  if  prolonged  to  the  limits  oi' 

the 


62 


TRIGONOMETRICAL  SURVEYING. 


the  apparent  celestial  sphere,  would  there  trace  the  circum- 
ference of  a great  circle,  which  would  be  the  meridian  of  that 
place.  All  the  points  of  the  earth’s  surface,  which  have  their 
zenith  in  that  circumference,  will  be  under  the  same  celestial 
meridian,  and  will  form  the  corresponding  terrestrial  meri- 
dian. If  the  earth  be  an  irregular  spheroid,  this  meridian  will 
be  a curve  of  double  curvature  ; but  if  the  earth  be  a solid  of 
revolution,  the  terrestrial  meridian  will  be  a plane  curve 

5.  If  the  earth  were  a sphere,  then  ever}'  point  upon  a 
terrestrial  meridian  would  be  at  an  equal  distance  from  the 
centre,  and  of  consequence  every  degree  upon  that  meridian 
would  be  of  equal  length.  But  if  the  earth  be  an  ellipsoid 
of  revolution  slightly  flattened  at  its  poles,  and  protuberant 
at  the  equator;  then,  as  will  be  shown  soon,  the  degrees  of 
the  terrestrial  meridian,  in  receding  from  the  equator  towards 
the  poles,  will  be  increased  in  the  duplicate  ratio  of  the  right 
sine  of  the  latitude  ; and  the  ratio  of  the  earth’s  axes,  as  well 
as  their  actual  magnitude,  may  be  ascertained  by  comparing 
the  lengths  of  a degree  on  the  meridian  in  different  latitudes. 
Hence  appears  the  great  importance  of  measuring  a degree. 

6.  Now,  instead  of  actually  tracing  a meridian  on  the  sur- 
face of  the  earth, — a measure  which  is  prevented  by  the  in- 
terposition of  mountains,  woods,  rivers,  and  seas, — a con- 
struction is  employed  which  furnishes  the  same  result.  It  con- 
sists in  this. 

Let  abcdef,  &c.  be  a series  of  triangles,  carried  on  as 
nearly  as  may  be,  in  the  direction  of  the  meridian,  according  to 


the  observations  in  art.  3.  These  triangles  are  really  spherical 
or  spheroidal  triangles  ; but  as  their  curvature  is  extremely 
small,  thej'  are  treated  the  same  as  rectilinear  triangles,  either 
by  reducing  them  to  the  chords  of  the  respective  terrestrial 
arcs  ac,  ab,  bc,  & c.  or  by  deducting  a third  ot  the  excess, 
of  the  sum  of  the  three  angles  of  each  triangle  above  two 
right  angles,  from  each  angle  of  that  triangle,  and  working 
with  the  remainders,  and  the  three  sides,  as  the  dimensions 
of  a plane  triangle  ; the  proper  reductions  to  the  centre  ot 
the  station,  to  the  horizon,  and  to  the  level  of  the  sea,  having 
been  previously  made.  These  computations  being  made 

throughout 


GEODESIC  OPERATIONS. 


63 


throughout  the  series,  the  sides  of  the  successive  triangles  are 
contemplated  as  arcs  of  the  terrestrial  spheriod.  Suppose 
that  we  knew,  by  observation,  and  the  computations  which 
will  be  explained  in  this  chapter,  the  azimuth , or  the  inclina- 
tion of  the  side  ac  to  the  first  portion  am  of  the  measured 
meridian,  and  that  we  find  by  trigonometry,  the  point  m where 
that  curve  will  cut  the  side  bc.  The  points  a,  b,  c,  being  in 
the  same  horizontal  plane,  the  line  am  will  also  be  in  that 
plane  : but,  because  of  the  curvature  of  the  earth,  the  pro- 
longation mm',  of  that  line,  will  be  found  above  the  plane  of  the 
second  horizontal  triangle  bcd  : if,  therefore,  without  chang- 
ing the  angle  cmm,  the  line  mm'  be  brought  down  to  coincide 
with  the  plane  of  this  second  triangle,  by  being  turned  about 
bc  as  an  axis,  the  point  m'  will  describe  an  arc  of  a circle, 
which  will  be  so  very  small,  that  it  may  be  regarded  as  a 
right  line  perpendicular  to  the  plane  bcd  : whence  it  follows, 
that  the  operation  is  reduced  to  bending  down  the  side  mm'  in 
the  plane  of  the  meridian,  and  calculating  the  distance  amm', 
to  find  the  position  of  the  point  m'.  By  bending  down  thus  in 
imagination,  one  after  another,  the  parts  of  the  meridian  on 
the  corresponding  horizontal  triangles,  we  may  obtain,  by  the 
aid  of  the  computation,  the  direction  and  the  length  of  such 
meridian,  from  one  extremity  of  the  series  of  triangles,  to  the 
other. 

A line  traced  in  the  manner  we  have  now  been  describing, 
or  deduced  from  trigonometrical  measures,  by  the  means  we 
have  indicated,  is  called  a geodetic  or  geodesic  line  ; it  has  the 
property  of  being  the  shortest  which  can  be  drawn  between 
its  two  extremities  on  the  surface  of  the  earth  ; and  it  is  there- 
fore the  proper  itinerary  measure  of  the  distance  between 
those  two  points.  Speaking  rigorously,  this  curve  differs  a 
little  from  the  terrestrial  meridian,  when  the  earth  is  not  a 
solid  of  revolution  : yet,  in  the  real  state  of  things,  the  dif- 
ference between  the  two  curves  is  so  extremely  minute,  that 
it  may  safely  be  disregarded. 

7.  If  now  we  conceive  a circle  perpendicular  to  the  celes- 
al  meridian,  and  passing  through  the  vertical  of  the  place 
of  the  observer,  it  will  represent  the  prime  vertical  of  that 
place.  The  series  of  all  the  points  of  the  earth’s  surface  which 
have  their  zenith  in  the  circumference  of  this  circle,  will  form 
the  perpendicular  to  the  meridian,  which  may  be  traced  in 
like  manner  as  the  meridian  itself.  ^ 

In  the  sphere  the  perpendiculars  to  the  meridian  are  great 
circles  which  all  intersect  mutually,  on  the  equator,  in  two 
points  diametrically  opposite  :.but  in  the  ellipsoid  of  revolu 

tion. 


64 


TRIGONOMETRICAL  SURVEYING. 


tion,  and  a fortiori  in  the  irregular  spheriod,  these  concurring 
perpendiculars  are  curves  of  double  curvature.  Whatever  be 
the  nature  of  the  terrestrial  spheriod,  the  parallels  to  the 
equator  are  curves  of  which  all  the  points  are  at  the  same 
latitude  : on  an  ellipsoid  of  revolution,  these  curves  are  plane 
and  circular. 

8.  The  situation  of  a place  is  determined,  when  we  know 
either  the  individual  perpendicular  to  .the  meridian,  or  the  in- 
dividual parallel  to  the  equator,  on  which  it  is  found,  and  its 
position  on  such  perpendicular,  or  on  such  parallel.  There- 
fore, when  all  the  triangles,  which  constitute  such  a series  as 
we  have  spoken  of.  have  been  computed,  according  to  the 
principles  just  sketched,  the  respective  positions  of  their  an- 
gular points,  either  by  means  of  their  longitudes  and  latitudes 
or  of  their  distances  from  the  first  meridian,  and  from  the  per- 
pendicular to  it.  The  following  is  the  method  of  computing 
these  distances. 

Suppose  that  the  triangles  abc,  bcd,  &c.  (see  the  fig.  to 
art-  6)  make  part  of  a chain  of  triangles,  of  which  the  sides 
are  arcs  of  great  circles  of  a sphere,  whose  radius  is  the  dis- 
tance from  the  level  or  surface  of  the  sea  to  the  centre  of  the 
earth  ; and  that  we  know  by  observation  the  angle  cax,  which 
measures  the  azimuth  of  the  side  ac,  or  its  inclination  to  the 
meridian  ax.  Then,  having  found  the  excess  e,  of  the  three 
angles  of  the  triangle  acc  (c c being  perpendicular  to  the  me- 
ridian) above  two  right  angles,  by  reason  of  a theorem  which 
will  be  demonstrated  in  prob.  8 of  this  chapter,  subtract  a third 
of  this  excess  from  each  angle  of  the  triangle,  and  thus,  by 
means  of  the  following  proportions  find  ac,  and  cc. 
sin  (90°  — |e  : cos  (cac  — |e)  : : ac  : ac  ; 

sin  (90° — ^e  : sin  (cac  — -^e)  : : ac  : c c.. 

The  azimuth  of  ab  is  known  immediately,  because  bax  = 
cab  — cax  ; and  if  the  spherical  excess  proper  to  the  triangle 
abm'  be  computed,  we  shall  have 

am'b  = 1 80'  — m'ab  — abm'  + e'. 

To  determine  the  sides  am',  bm(',  a third  of  e must  be  deducted 
from  each  of  the  angles  of  the  triangle  abm'  ; and  then  these 
proportions  will  obtain  : viz. 

sin  (180°  — m'ab  — abm'  + |e')  : sin  (abm’ — Ae')  : : ab  : am', 

sin  (180°  — m'ab  — abm'  + |-e')  : sin  (m'ab  — Ae)  : : ab  : bm'. 

In  each  of  the  right-angled  triangles  a6b,  m'Jd,  are  known 
tivo  angles  and  the  hypothenuse,  which  is  all  that  is  necessary 
to  determine  the  side§.  a b,  bh,  and  m 'd,  do.  Therefore  the  dis- 
tances of  the  points  e,  d,  from  the  meridian  and  from  the  per- 
pendicular, are  knowm. 


9.  Pro- 


GEODESIC  OPERATIONS. 


65 


t).  Proceeding  in  the  same  manner  with  the  triangle  acn,  or 
m'dn,  to  obtain  an  and  dn,  the  prolongation  of  cd  ; and 
then  with  the  triangle  dnf  to  find  the  side  nf  and  the  angles 
dnf  dfn,  it  will  be  easy  to  calculate  the  rectangular  co-ordi* 
nates  of  the  point  f. 

The  distance /f  and  the  angles  dfn,  nf/1,  being  thus  known, 
we  shall  hare  (th.  6 cor.  3 Geom.) 

/fP  = 180°  EFD  DFN  — NF/. 

So  that,  in  the  right  angled  triangle  /fp,  two  angles  and  one 
side  are  known  ; and  therefore  the  appropriate  spherical  ex- 
cess may  be  computed,  and  thence  the  angle  fp f and  the  sides 
fp,  fp.  Resolving  next  the  right-angled  triangle  pep,  we  shall 
in  like  manner  obtain  the  position  of  the  point  e with  respect 
to  the  meridian  ax,  and  to  its  perpendicular  ay  ; that  is  to  say, 
the  distances  Ee,  and  \e  — ap  — cf.  And  thus  may  the  computist 
proceed  through  the  whole  of  the  series.  It  is  requisite  how- 
ever, previous  to  these  calculations,  to  draw,  by  any  suitable 
scale,  the  chain  of  triangles  observed,  in  order  to  see  whether 
any  of  the  subsidiary  triangles  acn,  nfp,  &c.  formed  to  faci- 
litate the  computation  of  the  distances  from  the  meridian,  and 
from  the  perpendicular  to  it,  are  too  obtuse  or  too  acute. 

Such,  in  few  words,  is  the  method  to  be  followed,  when  we 
have  principally  in  view  the  finding  the  length  of  the  portion 
of  the  meridian  comprised  bfttweea  any  two  points,  as  a and 
x.  It  is  obvious  that,  in  the  course  of  the  computations,  the 
azimurbs  of  a great  number  of  the  sides  of  triangles  in  the 
series  is  determined  ; it  will  be  easy  therefore  to  check  and 
verify  the  work  in  its  pro  ;ess,  by  comparing  the  azimuths 
found  by  observation,  with  those  resulting  from  the  calcula- 
tions, The  amplitude  of  the  whole  arc  of  the  meridian 
measured,  is  found  by  ascertaining  the  latitude  at  each  of  its 
extremities  ; that  is,  commonly  by  finding  the  differences  of 
the  zenith  distances  of  some  known  fixed  star,  at  both  those 
extremities 

10  Some  mathematicians,  employed  in  this  kind  of  opera- 
tions, have  adopted  different  means  from  the  above.  They 
draw  through  the  summits  of  all  the  triangles,  parallels  to  the 
meridian  and  to  its  perpendicular  ; by  these  means,  the  sides 
of  the  triangles  become  the  hypothenuses  of  right-angled  tri- 
angles, which  they  compute  in  order,  proceeding  from  some 
known  azimuth,  and  without  regarding  the  spherical  excess, 
considering  all  the  triangles  of  the  chain  as  described  on  a 
plane  surface.  This  method,  however,  is  manifestly  defective 
in  point  of  accuracy 

Others  have  computed  the  sides  and  angles  of  all  the  tri- 
angles, by  the  rules  of  spherical  trigonometry.  Otners  again, 

Voj,  If,  10  reduce 


TRIGONOMETRICAL  SURVEYING. 


$6 

reduce  the  observed  angles  to  angles  of  the  chords  of  the  re- 
spective arches,  and  calculate  by  plane  trigonometry,  from 
such  reduced  angles  and  their  chords.  Either  of  these  two 
methods  is  equally  correct  as  that  by  means  of  the  spherical 
excess  : so  that  the  principal  reason  for  preferring  one  of 
these  to  the  other,  must  be  derived  from  its  relative  facility. 
As  to  the  methods  in  which  the  several  triangles  are  contem- 
plated as  spheroidal,  they  are  abstruse  and  difficult,  and  may, 
happily,  be  safely  disregarded  : for  M Lengendre  has  demon- 
strated in  Meinoires  de  la  Classe  des  Sciences  Physiques  et  Alathe- 
matiques  de  l'  Institute  1806,  pa.  130,  that  the  difference  be- 
tween spherical  and  spheroidal  angles,  is  less  than  one  sixtieth 
of  a second,  in  the  the  greatest  of  the  triangles  which  occur- 
red in  the  late  measurement  of  an  arc  of  a meridian,  between 
the  parallels  of  Dunkirk  and  Barcelona. 

11.  Trigonometrical  surveys  for  the  purpose  of  measuring 
a degree  of  a meridian  in  different  latitudes,  and  thence  in- 
ferring the  figure  of  the  earth,  have  been  undertaken  by  dif- 
ferent philosophers,  under  the  patronage  of  different  go- 
vernments. As  by  M.  Mapertuis,  Clairaut,  Sic.  in  Lapland, 
1736  : by  M.  Bouguer  and  Condamine,  at  the  equator,  1736 — 
1743  ; by  Cassini,  in  lat.  45°,  1739 — 40  ; by  Boscovich  and 
Lemaire,  lat.  43°,  1752  ; by  Beccaria,  lat  44°  44 , 1768  ; by 
Mason  and  Dixon  in  America,  1764 — 8 ; by  Major  Lambton, 
in  the  East  Indies,  1803  ; by  Mechain,  Delambre,  &c.  France, 
&c  1790—1805  ; by  Swanberg,  Ofverbom,  &c.  in  Lapland, 
1802  ; and  by  General  Roy,  Colonel  Williams,  Mr.  Dalby,  and 
Colonel  Mudge,  in  England,  from  1784  to  the  present  time. 
The  three  last  mentioned  of  these  surveys  are  doubtless  the 
most  accurate  and  important. 

The  trigonometrical  survey  in  England  was  first  commenced, 
in  conjunction  with  similar  operations  in  France,  in  order  to 
determine  the  difference  of  longitude  between  the  meridians 
of  the  Greenwich  and  Paris  observatories  ; for  this  purpose, 
three  of  the  French  Academecians,  M.  M.  Cassini,  Mechain, 
and  Legendre,  met  General  Roy  and  Dr.  (now  Sir  Charles) 
Blagden,  at  Dover,  to  adjust  their  plans  of  operation.  In  the 
course  of  the  survey,  however,  the  English  philosophers,  se- 
lected from  the  Royal  Artillery  officers,  expanded  their  views, 
and  pursued  their  operations,  under  the  patronage,  and  at 
the  expence  of  the  Honourable  Board  of  Ordnance,  in  order 
to  perfect  the  geography  of  England,  and  to  determine  the 
lengths  of  as  many  degrees  on  the  meridian  as  fell  within  the 
compass  of  their  labours. 

12.  It  is  not  our  province  to  enter  into  the  history  of  these 

surveys 


GEODESIC  OPERATIONS. 


67 


surveys  : but  it  may  be  interesting  and  instructive  to  speak  a 
little  of  the  instruments  employed,  and  of  the  extreme  accu- 
racy of  some  of  the  results  obtained  by  them. 

These  instruments  are,  besides  the  signals,  those  for  mea- 
suring distances,  and  those  for  measuring  angles.  The  French 
philosophers  used  for  the  former  purpose,  in  their  measure- 
ment to  determine  the  length  of  the  metre,  rulers  of  platina 
and  of  copper,  forming  metalic  thermometers.  The  Swedish 
mathematicians,  Swanberg  and  Ofverbom,  employed  iron 
bars,  covered  towards  each  extremity  with  plates  of  silver. 
General  Roy  commenced  his  measurement  of  the  base  at 
Hounslow  Heath  with  deal  rods,  each  of  20  feet  in  length. 
Though  they,  however,  were  made  of  the  best  seasoned  tim- 
ber, were  perfectly  straight,  and  were  secured  from  bending 
in  the  most  effectual  manner  ; yet  the  changes  in  their  lengths, 
occasioned  by  the  variable  moisture  and  dryness  of  the  air? 
were  so  great,  as  to  take  away  all  confidence  in  the  results 
deduced  from  them.  Afterwards,  in  consequence  of  having 
found  by  experiments,  that  a solid  bar  of  glass  is  more  dilata- 
ble than  a tube  of  the  same  matter,  glass  tubes  were  substi- 
tuted for  the  deal  rods  They  were  each  20  feet  long,  inclosed 
in  wooden  frames,  so  as  to  allow  only  of  expansion  or  con- 
traction in  length,  from  heat  or  cold,  according  to  a law 
ascertained  by  experiments.  The  base  measured  with  these 
was  found  to  be  27404-08  feet,  or  about  5-19  miles.  Several 
years  afterwards  the  same  base  was  remeasured  by  Colonel 
Mudge,  with  a steel-chain  of  100  feet  long,  constructed  by 
Ramsden,  and  jointed  somewhat  like  a watch-chain.  This 
chain  was  always  stretched  to  the  same  tension,  supported  on 
troughs  laid  horizontally,  and  allowances  were  made  for 
changes  in  its  length  by  reason  of  variations  of  temperature,  at 
the  rate  of  -0075  of  an  inch  for  each  degree  of  heat  from  62° 
of  Fahrenheit  : the  result  of  the  measurement  by  this  chain  was 
found  not  to  differ  more  than  2f  inches,  from  General  Roy’s 
determination  by  means  of  the  glass  tubes  : a minute  differ- 
ence in  a distance  of  more  than  5 miles  ; which,  considering 
that  the  measurements  were  effected  by  different  persons,  and 
with  different  instruments,  is  a remarkable  confirmation  of  the 
accuracy  of  both  operations.  And  further,  as  steel  chains  can 
be  used  with  more  facility  and  convenience  than  glass  rods, 
this  remeasurement  determines  the  question  of  the  compara- 
tive fitness  of  these  two  kinds  of  instruments. 

13.  For  the  determination  of  angles,  the  French  and  Swe- 
dish philosophers  employed  repeating  circles  of  Borda’s  con- 
struction : instruments  which  are  extremely  portable,  and  with 
which,  though  they  are  not  above  14  inches  in  diameter,  the 

observers 


68 


TRIGONOMETRICAL  SURVEYING. 


observers  can  take  angles  to  within  1"  or  2'  of  the  troth 
But  this  kind  of  instrument,  however  great  its  ingenuity  in 
theory,  has  the  accuracy  of  its  observations  necessarily  limited 
by  the  imperfections  of  the  small  telescope  which  must  be 
attached  to  it  General  Roy  and  Colonel  Mudge  made  use  of 
a very  excellent  theodolite  constructed  by  Ramsden,  which, 
having  both  an  altitude  and  an  azimuth  circle,  combines  the 
powers  of  a theodolite,  a quadrant,  and  a transit  instrument, 
and  is  capable  of  measuring  horizontal  angles  to  fractions  of  a 
second.  This  instrument,  besides,  has  a telescope  of  a much 
higher  magnifying  power  than  had  ever  before  been  applied 
to  observations  purely  terrestrial  ; and  this  is  one  of  the  supe- 
riorities  in  its  construction,  to  which  is  to  be  ascribed  the  ex 
treme  accuracy  in  the  results  of  this  trigonometrical  survey. 

Another  circumstance  which  has  augmented  the  accuracy 
of  the  English  measures,  arises  from  the  mode  of  fixing  and 
using  this  theodolite.  In  the  method  pursued  by  the  Con- 
tinental mathematicians,  a reduction  is  necessary  to  the  plane 
of  the  horizon,  and  another  to  bring  the  observed  angles  to 
the  true  angles  at  the  centres  of  the  signals  : these  reductions, 
of  course,  require  formulas  of  computation,  the  actual  em- 
ployment of  which  may  lead  to  error.  But.  in  the  trigono- 
metrical survey  of  England,  great  care  has  always  been  taken 
to  place  the  centre  of  the  theodolite  exactly  in  the  vertical  line, 
previously  or  subsequently  occupied  by  the  centre  of  the 
signal  : the  theodolite  is  also  placed  in  a perfectly  horizontal 
position.  Indeed,  as  has  been  observed  by  a competent  judge, 
“ In  no  other  survey  has  the  work  in  the  field  been  conducted 
so  much  with  a view  to  save  that  in  the  closet,  and  at  the  same 
time  to  avoid  all  those  causes  of  error,  however  minute,  that 
are  not  essentially  involved  in  the  nature  of  the  problem.  The 
French  mathematicians  trust  to  the  correction  of  those  errors  . 
the  English  endeavour  to  cut  them  off" entirely  ; and  it  can 
hardly  be  doubted  that  the  latter,  though  perhaps  the  slower 
and  more  expensive,  is  by  far  the  safest  proceeding.” 

14.  In  proof  of  the  great  correctness  of  the  English  survey, 
we  shall  state  a very  few  particulars,  besides  what  is  already 
mentioned  in  art.  12. 

General  Roy,  who  first  measured  the  base  on  Hounslow 
Heath,  measured  another  on  the  flat  ground  of  Romney- 
Marsh  in  Kent,  near  the  southern  extremity  of  the  first  series 
of  triangles,  and  at  the  distance  of  more  than  60  miles  from 
the  first  base.  The  length  of  this  base  of  verification,  as 
actually  measured,  compared  with  that  resulting  from  the 
computation  through  the  whole  series  of  triangles,  diflerec' 
only  by  28  inches. 


Colonel 


GEODESIC  OPERATIONS. 


69 


Colonel  Mudge  measured  another  base  of  verification  on 
Salisbury  plain.  Its  length  was  36574-4  feet,  or  more  than 
7 miles  ; the  measurement  did  not  differ  more  than  one  inch 
from  the  computation  carried  through  the  series  of  triangles 
from  Hounslow  Heath  to  Salisbury  Plain.  A most  remarkable 
proof  of  the  accuracy  with  which  all  the  angles,  as  well  as 
the  two  bases,  were  measured  ! 

The  distance  between  Beachy-Head  in  Sussex,  and  Dun- 
nose  in  the  Isle  of  Wight,  as  deduced  from  a mean  of  four 
series  of  triangles,  is  339397  feet  or  more  than  64i  miles. 
The  extremes  of  the  four  determinations  do  not  differ  more 
than  7 feet,  which  is  less  than  1|  inches  in  a mile.  Instances 
of  this  kind  frequently  occur  in  the  English  survey*1.  But  we 
have  not  room  to  specify  more.  We  must  now  proceed  to 
discuss  the  most  important  problems  connected  with  this  sub- 
ject ; and  refer  those  who  are  desirous  to  consider  it  more 
minutely,  to  Colonel  Sludge’s,  “ Account  of  the  Trigonome- 
trical Survey  Mechain  and  Delambre,  “ Base  du  Systeme 
Metrique  Decimal  Swanberg,  “ Exposition  des  Operations 
faitesen  Lapponie  and  Puissant’s  works  entitled  “ Geode- 
sie”  and  “ Traite  de  Topographie,  d’  Arpentage,  &c.” 


SECTION  II. 

Problems  connected  with  the  detail  of  Operations  in  Extensive 
Trigonometrical  Surveys. 

PROBLEM  I. 

It  is  required  to  determine  the  Most  Advantageous 
Conditions  of  Triangles. 


1.  In  any  rectilinear  triangle  abc,  it  is  from  the  propot 
fionality  of  sides  to  the  sines  of  their  opposite  angles,  ab 
bc  : : sin  c : sin  a,  and  consequently  ab.  sin  ^ 

a = bc  . sin  c.  Let  ab  be  the  base,  which 
is  supposed  to  be  measured  without  percep- 
tible error,  and  .which  therefore  is  assumed 
as  constant ; then  finding  the  extremely  j<^~ 


* Puissant,  in  his  “ Geod^sie,”  after  quoting  some  of  them,  says, 
‘‘  Neanmoins,  jusqu’a  present,  nenn’egale  en  exactitude  les  opera- 
tions geodesiques  qui  out  servi  de  foil  dement  a notre  systeme  metri- 
que-”  He,  however,  gives  no  instances.  We  have  no  wish  to  depre- 
ciate the  labours  of  the  French  measurers  : but  we  cannot  yield  them 
the  preference  on  mere  assertion - 


small 


70 


TRIGONOMETRICAL  SURVEYING. 


small  variation  or  fluxion  of  the  equation  on  this  hypothesis, 
it  is  ab  . cos  a . a = sin  c . bc  + bc  . cos  c . c.  Here,  since 
we  are  ignorant  of  the  magnitude  of  the  errors  or  variations 
expressed  by  a and  c,  suppose  them  to  be  equal  (a  probable 
supposition,  as  they  are  both  taken  by  the  same  instrument), 
and  each  denoted  by  v : then  will 


BC  =1)  X 


AB  COS  A—  BC  COS  C 
sin  c 


or,  substituting-^—  for  its  equal the  equation  will  be- 
sin  a ’ sin  c 

• . . , cos  a cos  c , 

come  bc  = v X (bc  . bc  . ) ; 

. sin  a sin  c 

or,  finally  bc  = v bc  (cot  a — cot  c). 

This  equation  (in  the  use  of  which  it  must  be  recollected 
that  v taken  in  seconds  should  be  divided  by  it",  that  is  by 
the  length  of  the  radius  expressed  in  seconds)  gives  the  error 
bc  in  the  estimation  of  bc  occasioned  by  the  errors  in  the 
angles  a and  c.  Hence,  that  these  errors,  supposing  them  to 
be  equal,  may  have  no  influence  on  the  determination  of  bc, 
we  must  have  a = c,  for  in  that  case  the  second  member  of  the 
equation  will  vanish. 

2.  But,  as  the  two  errors,  denoted  by  a,  and  c,  which  we 
have  supposed  to  be  of  the  same  kind,  or  in  the  same  direc- 
tion, may  be  committed  in  different  directions,  when  the  equa- 
tion will  be  bc  ==  it  v . bc  (cot  a -f-  cot  c)  ; we  must  enquire 
what  magnitude  the  angles  a and  c ought  to  have,  so  that  the 
sum  of  their  cotangents  shall  have  the  least  value  possible  ; 


for  in  this  state  it  is  manifest  that  bc  will  have  its  least  value 
But,  by  the  formulae  in  chap.  3,  we  have 

sin  (A  + c) sin  (a  + C)  _ 


cot  (a  -f  c) 


. sin  c 
2 sin  b 


icos(Acc  c)  — £cos(a-J-  c) 


cos  (a  73  c)  -f-  cos  b’ 

• . 2 sin  b 

, bc  = ± v . bc  . . 

cos(a  it.  c)  + cos  b 
And  hence,  whatever  be  the  magnitude  of  the  angle  e,  the 
error  in  the  value  of  bc  will  be  the  least  when  cos  (aod  c)  is 
the  greatest  possible,  which  is  when  a = c. 

We  may  therefore  infer,  for  a general  rule,  that  the  most 
advantageous  state  of  a triangle,  when  we  would  determine  one 
side  only,  is  when  the  base  is  equal  to  the  side  sought. 

3.  Since,  by  this  rule,  the  base  should  be  equal  to  the  side 
sought,  it  is  evident  that  when  we  would  determine  two  sides, 
the  most  advantageous  condition  of  a triangle  is  that  it  be  equi- 
lateral. 


Consequently 


4.  It 


GEODESIC  OPERATIONS. 


71 


4.  It  rarely  happens,  however,  that  a base  can  be  commo- 
diously  measured  which  is  as  long  as  the  sides  sought.  Sup- 
posing, therefore,  that  the  length  of  the  base  is  limited,  but 
that  its  direction  at  least  may  be  chosen  at  pleasure,  we  proceed 
to  enquire  what  that  direction  should  be,  in  the  case  where 
one  only  of  the  other  two  sides  of  the  triangles  is  to  be  deter- 
mined. 

Let  it  be  imagined,  as  before,  that  ab  is  the  base  of  the  tri- 
angle abc,  and  bc  the  side  required  It  is  proposed  to  find 
the  least  value  of  cot  a q=  cot  c,  when  we  cannot  have  a = c. 

Now,  in  tbe  case  where  the  negative  sign  obtains,  we  have 

AB— BC  COS  B BC  — AB  . COS  B AB2  — BC2 

COt  A — COt  C = : — = - 

BC  • Sin  B AB  . sin  B AB  . BC  • Sin  B‘ 

This  equation  again  manifestly  indicates  the  equality  of  ab  and 
bc,  in  circumstances  where  it  is  possible  : but  if  ab  and  bc  are 
constant,  it  is  evident,  from  the  form  of  the  denominator  of 
the  last  fraction,  that  the  fraction  itself  will  be  the  least,  or 
cot  a— cot  c the  least,  when  sin  b is  a maximum,  that  is,  when 
b = 90°. 


5.  When  the  positive  sign  obtains,  we  have  cot  a -J-  cot  c= 


cot  a 4 


v/  (BC2  — AB2  sin2  a) 


COt  A -f"  \/  (- 


T-1)- 


ab  sin  a ‘ v 'ab2  sin  2 a 

Here,  the  least  value  of  the  expression  under  the  radical  sign, 
is  obviously  when  a = 90°.  And  in  that  case  the  first  term, 
cot  a,  would  disappear.  Therefore  the  least  value  of  cot  a 4* 
cot  c,  obtains  when  a = 90°  ; conformably  to  the  rule  given 
by  M.  Bouguer  {Fig.  de  la  Terre,  pa.  88).  But  we  have 
already  seen  that  in  the  case  of  cot  a— cot  c,  we  must  have 
b = 90.  Whence  we  conclude,  since  the  conditions  a = 90°, 
b = 90°,  cannot  obtain  simultaneously,  that  a medium  result 
would  give  a = b. 

If  we  apply  to  the  side  ac  the  same  reasoning  as  to  bc, 
similar  results  will  be  obtained  : therefore  in  general,  when 
the  base  cannot  be  equal  to  one  or  to  both  the  sides  required,  the 
most  advantageous  condition  of  the  triangle  is,  that  the  base  be 
the  longest  possible,  and  that  the  two  angles  at  the  base  be  equal. 
These  equal  angles  however,  should  never,  if  possible,  be 
less  than  23  degrees. 


PROBLEM  II. 


To  deduce,  from  Angles  measured  Out  of  one  of  the  stations, 
but  Near  it,  the  True  Angles  at  the  station. 

When  the  centre  of  the  instrument  cannot  be  placed  in  the 
vertical  line  occupied  by  the  axis  of  a signal,  the  angles  ob- 
served must  undergo  a reduction,  according  to  circumstances. 

1.  Let 


72 


TRIGONOMETRICAL  SURVEYING. 


1.  Let  c be  the  centre  of  the  station, 
p the  place  of  the  centre  of  the  instru- 
ment, or  the  summit  of  the  observed  an- 
gle apb  : it  is  required  to  find  c,  the 
measure  of  acf.  supposing  there  to  be 
known  apb  — r,  bpc  = p,  cp  = d,  bc 

= L,  AC  — R. 

Since  the  exterior  angle  of  a triangle  is  equal  to  the  sum 
of  the  two  interior  opposite  angles  (th  16  Geom.),  we  have, 
with  respect  to  the  triangle  iap,  aib  = p -j-  iap  ; and  with 
regard  to  the  triangle  bio,  aib  = c + cbp.  Making  these 
two  values  of  aib  equal,  and  transposing  iap,  there  results 
c = p + iap  — CBP. 

But  the  triangles  cap,  cbp,  give 

cp  . d • sin  ' p-f-fi) 

sin  cap  = sin  iap  = — sin  apc  = . 

AC  R 


sin  cbp  =- 


J . sin  p 

sin  bpc  = 


BC  L 

And,  as  the  angles  £ap,  cbp,  are,  by  the  hypothesis  of  the 
problem,  always  very  small,  their  sines  may  be  substituted 
for  their  arcs  or  measures  : therefore 

d sin  (p-f /,)  d . sin  p 


R 1 

Or,  to  have  the  reduction  in  seconds, 

d , sin  (p-l.fi') . sin  6. 

c — p = — — ). 

sinl  ' u l 1 

The  use  of  this  formula  cannot  in  any  case  be  embarrassing, 
provided  the  signs  of  sin  p,  and  sin  (p  + p ) be  attended  to. 
Thus,  the  first  term  of  the  correction  will  be  positive,  if  the 
angle  (p  + p)  is  comprised  between  0 and  180°  ; and  it  will 
become  negative,  if  that  angle  surpass  180°.  The  contrary 
will  obtain  in  the  same  circumstances  with  regard  to  the  se- 
cond term,  which  answers  to' the  angle  of  direction  p.  The 
letter  r denotes  the  distance  of  the  object  a to  the  right,  r 
the  distance  of  the  object  b situated  to  the  left,  and  p the 
angle  at  the  place  of  observation,  between  the  centre  of  the 
station  and  the  object  to  the  left. 


2.  An  approximate  reduction  to  the  centre  may  indeed  be 
obtained  by  a single  term  : but  it  is  not  quite  so  correct  as  the 
form  above.  For,  by  reducing  the  two  fractions  in  the  second 
member  of  the  last  equation  but  one  to  a common  denominator, 
the  correction  becomes 

d l . sin  (p-f  fi)-  dp.  ■ sin  p 
c — P . 

J.R 

1 • _ R • si'1  A H . sin  a 

But  the  triangle  abc  gives  r.  = — : =- — 

& & sin  b sin  (A-f  c) 

And 


GEODESIC  OPERATIONS. 


73 


R bin  A 


And  because  p is  always  very  nearly  equal  to  c,  the  sine  of 
a + p will  differ  extremely  little  from  sin  (a  + c),  and  may 

therefore  be  substituted  for  it,  making  l = 

Hence  we  manifestly  have 

c — p 


d ■ si- 


sin  (a+p)' 
(p  r p ) — d .sin  p . sin  (a+p) 


Which,  by  taking  the  expanded  expressions,  for  sin  (p+/>)> 
and  sin  (a+p,)  and  reducing  to  seconds,  gives 
d sin  p • sin  (a  — fi) 

C — P = — : • 

sin  1 R sin  A 

3.  When  either  of  the  distances  u,  l,  becomes  infinite,  with 
respect  to  d,  the  corresponding  term  in  the  expression  art.  1 
of  this  problem,  vanishes,  and  we  have  accordingly 
d ■ sin  p ^ d . sin  (p  + *>) 


s n 1 


or  c 


R sin  1 


The  first  of  these  will  apply  when  the  object  a is  a heavenly 
body,  the  second  when  b is  one.  When  both  a and  b are 
such,  then  c — p = 0. 

But  without  supposing  either  a or  b infinite,  we  may  have 
c — p = 0,  or  c = p in  innumerable  instances  : that  is,  in  all 
cases  in  which  the  centre  p of  the  instrument  is  placed  in  the 
circumference  of  the  circle  that  passes  through  the  three 
points  a,  b,  c ; or  when  the  angle  bpc  is  equal  to  the  angle 
bac,  or  to  bac  + 180°.  Whence,  though  c should  be  inac- 
cessible, the  angle  acb  may  commonly  be  obtained  by  obser- 
vation, without  any  computation.  It  may  further  be  observed, 
that  when  p falls  iu  the  circumference  of  the  circle  passing 
through  the  three  points  a,  b,  c,  the  angles  a,  b,  c,  may  be 
determined  solely  by  measuring  the  angles  apb  and  bpc.  For, 
the  opposite  angles  abc,  apc,  of  the  quadrangle  inscribed  in 
a circle,  are  (theor.  54  Geom.)  = 180°  Consequently,  abc 
= 1 {30°  — apc,  add  bac  = 180°  — (abc+acb)  — 180°  — 
(abc  + apb). 

4.  If  one  of  the  objects,  viewed  from  a further  station,  be 
a vane  or  staff  in  the  centre  of  a steeple,  it  will  frequently 
happen  that  such  object,  when  the  observer  comes  near  it,  is 
both  invisible  and  inaccessible.  Still  there  are  various  me- 
thods of  finding  the  exact  angle  at  c.  Suppose,  for  example, 
the  signal  staff  be  in  the  centre  of  a 
circular  tower,  and  that  the  angle  apb 
was  taken  at  p near  its  base.  Let  the 
tangents  pt,  pt’,  be  marked,  and  on 
them  two  equal  and  arbitrary  distances 
pm,  pm , be  measured.  Bisect  mm  at 
the  point  n ; and,  placing  there  a sigoal- 

Voti.  II.  11 


staff 


14 


TRIGONOMETRICAL  SURVEYING. 


staff,  measure  the  angle  titb,  which,  (since  pn  prolonged  ob- 
viously passes  through  c the  centre,)  will  be  the  angle  p of 
the  preceding  investigation.  Also,  the  distance  ps  added  to 
the  radius  c s of  the  tower,  will  give  pc  = d in  the  former  in- 
vestigation. 


If  the  circumference  of  the  tower  cannot  be  measured,  and 
tiie  radius  thence  inferred,  proceed  thus  : Measure  the  angles 
BFT,  bpt',  then  will  epc  = i (bpt  + bpt')  = p ; and  cpt  = 
bpt  — bpc  : Measure  pt,  then  pc  = pt  . sec  cpt  = d.  "With 
the  values  of  p and  d,  thus  obtained,  proceed  as  before 

5.  If  the  base  of  the  tower  be  polygonal  and  regular,  as 
most  commonly  happens  ; assume  p in  the  point  of  intersec- 
tion of  two  of  the  sides  prolonged,  and  bpc  = 4 (bpt+bpt) 
as  before,  pt  = the  distance  from  p to 
the  middle  of  one  of  the  sides  whose 
prolongation  passes  through  p ; and 
hence  pc  is  found,  as  above.  If  the 
figure  be  a regular  hexagon,  then  the 
triangle  p mm,  is  equilateral,  and  pc 
— trim  y/  3. 


PROBLEM  mu 

* 

To  Reduce  Angles  measured  in  a Plane  Inclined  to  the 
Horizon,  to  the  Corresponding  Angles  in  the  Horizontal 
Plane. 


Let  bca  be  an  angle  measured  in  a plane  inclined  to  the 
horizon,  and  let  b ca'  be  the  corresponding  angle  in  the  ho- 
rizontal plane.  Let  d and  Id'  be  the  zenith  distances,  or  the 
complements  of  the  angles  of  elevation  aca',  bcb'.  Then 
from  z the  zenith  of  the  observer, 
or  of  the  angle  c,  draw  the  arcs  za, 
zb,  of  vertical  circles,  measuring  the 
zenith  distances  d,  d',  and  draw  the 
arc  ab  of  another  great  circle  to 
measure  the  angle  c.  It  follows 
from  this  construction,  that  the  an- 
gle z,  of  the  spherical  triangle  zab, 
is  equal  to  the  horizontal  angle 
a'cb'  ; and  that,  to  find  it,  the  three  sides  za  = d,  zb  = d\ 
ab  = c,  are  given.  Call  the  sum  of  these  s ; then  the  result- 
ing formulae  of  prob.  2 ch.  iv,  applied  to  the  present  instance, 
becomes 


sin  %z 


sin  ic  s»=  y/ 


sin  3(5  — c/)  . sin  jj(s-d') 
sin  d • sin  d' 


If 


GEODESIC  OPERATIONS. 


75 


If  h and  V represent  the  angles  of  altitude  aca'  bcb',  the 
preceding  expression  will  become 

s;n  -J-  h — n)  . sin  i(c  -f-  h'  — h) 

sin  lc  = y 7 I • 

2 cos  h ci>s  k 

Or,  in  logarithms, 

log  sin  ic  = 4(20  log  sin  -|(c  + h — h‘)  + log  sin. 
i(c  4*  K — h)  — log  cos  h — log  cos  h'). 

Car.  1.  If  h = h',  then  is  sin  4c  = h n 1 AC-  . an(j 

2 COS  h 


log  sin  Aa'cb'  = 10  + log  sin  Lacb  — log  cos  h. 

Cor.  2.  If  the  angles  h and  K be  very  small,  and  nearly 
equal  ; then,  since  the  cosines  of  small  angles  vary  extremely 
slowly,  we  may,  without  sinsible  error,  take 
log  sin  4a'cb'=  10  + log  sin  Aacb  — log  cos  \{h  + h'). 

Cor.  3.  In  this  case  the  corrections  = a'cb'—  acb,  may 
fee  found  by  the  expression 

x — sin  l"(tan4c(40  — d-—~)2  —cot 


And  in  this  formula,  as  well  as  the  first  given  for  sin  \c,  d and 
d'  may  be  either  one  or  both  greater  or  less  than  a quadrant ; 
that  is,  the  equations  will  obtain  whether  aca'  and  bcb'  be  each 
an  elevation  or  a depression. 


Scholium.  By  means  of  this  problem,  if  the  altitude  of  a 
hill  be  found  barometrically,  according  to  the  method  describ- 
ed in  the  1st  volume  or  geometrically  according  to  some  of 
those  described  in  heights  and  distances,  or  that  given  in  the 
following  problem  ; then,  finding  the  angles  formed  at  the 
place  of  observation,  by  any  objects  in  the  country  below,  and 
their  respective  angles  of  depression,  their  horizontal  angles, 
and  thence  their  distances  may  be  found,  and  their  relative  » 
places  fixed  in  a map  of  the  country  ; taking  care  to  have  a 
sufficient  number  of  angles  between  intersecting  lines,  to  verify 
the  operations. 


PROBLEM  IT. 


Given  the  Angles  of  Elevation  of  Any  Distant  object,  taken 
at  Three  places  in  a Horizontal  Right  Line,  which  does 
not  pass  through  the  point  directly  below  the  object  ; and 
the  Respective  Distances  between  the  stations  ; to  find  the 
Height  of  the  Object,  and  its  Distance  from  either  station. 
Let  aed  be  the  horizontal  plane  : fe  the  perpendicular 
height  of  the  object  f above  that  plane  ; a,  b,  c,  the  three 
places  of  observation  ; fae,  fbe,  fce,  the  respective  angles 

ot 


76 


TRIGONOMETRICAL  SURVEYING. 


of  elevation,  and  ab,  bc,  the 
given  distances.  Then,  since 
the  triangles  aef,  bef,  cef, 
are  all  right  angled  at  e,  the  dis- 
tances ae,  be,  ce,  will  mani- 
festly be  as  the  cotangents  of 
the  angles  of  elevation  at  a,  b, 
and  c : and  we  have  to  deter- 
mine the  point  e,  so  that  those 
lines  may  have  that  ratio.  To 
effect  this  geometrically  use  the  following 

Construction.  Take  bm,  on  ac  produced,  equal  to  ec,  bn 
equal  to  ab  ; and  make 

mg  : bm  (—  bc)  : : cot  a : cot  b, 
and  bn(  = ab)  : ng  : : cot  b : cot  c. 

With  the  lines  mn,  mg,  ng,  constitute  the  triangle  mng  ; and 
join  bg.  Draw  ae  so,  that  the  angle  eab  may  be  equal  to  mcb  ; 
this  line  will  meet  bg  produced  in  e.  the  point  in  the  horizon- 
tal plane  falling  perpendicularly  below  f. 

Demonstration.  By  the  similar  triangles  aeb,  gmb,  we 
have  ae  : be  : : mg  : mb  : : cot  A : cot  b, 
and  be  : ba  (=  bn)  : : bm  : bg. 

Therefore  the  triangles  bec,  bgn,  are  similar  ; consequently 
be  : ec  : : bn  : ng  : : cot  b : cot  c.  Whence  it  is  obvious  that 
ae,  be.  ce,  are  respectively  as  cot  a,  cot  b,  cot  c. 

Calculation.  In  the  triangle  mgn,  all  the  sides  are  given,  to 
find  the  angle  gmn  = angle  aeb.  Then,  in  the  triangle  mob, 
two  sides  and  the  included  angle  are  given,  to  find  the  angle 
mgb  = angle  eab.  Hence,  in  the  triangle  aeb,  are  known  ab 
and  all  the  angles,  to  find  ae,  and  be.  And  then  ef  = ae  . tan 
a = be  . tan  B. 

Otherwise,  independent  of  the  construction,  thus. 

Put  ab  = d,  bc  = d,  ef  = x ; and  then  express  algebraic- 
ally the  following  theorem,  given  at  p.  128  Simpson's  Select 
Exercises  : 

AE2  . BC  -f-  CE2  . AB  = BE2  . AC  -f-  AC  . AB  . BC, 

the  line  eb  being  drawn  from  the  vertex  e of  the  triangle  ace, 
to  any  point  b in  the  base.  The  equation  thence  originating  is 
dx2  . cot2  A-f-DX2  . cot2  c = (D-ft/)x2  . cot2  b -f-  (d  + ^)  d d. 
And  from  this,  by  transposing  all  the  unknown  terms  to  one 
side,  and  extracting  the  root,  their  results 

x _ . (D  + c>Drf . 

^ d , cot2  A -f-  D . COt2  C — ( D + d)  cot  2 B 

Whence 


GEODESIC  OPERATIONS. 


77 


Whence  ef  is  known,  and  the  distances  ae,  be,  ce,  are  readily 
found. 

Cor  When  d = d,  or  d -f-  d = 2d  = 2d,  the  expression 
becomes  better  suited  foi  logarithmic  computation,  being  then 
x = d = y/  (A  cot2  a + i cot2  c — cot2  b). 

In  this  case,  therefore,  the  rule  is  as  follows  : Double  the  log. 
cotangents  of  the  angles  of  elevation  of  the  extreme  stations, 
find  the  natural  numbers  answering  thereto,  and  take  half 
their  sum  ; from  which  subtract  the  natural  number  answer- 
ing to  twice  the  log  cotangent  of  the  middle  angle  of  eleva- 
tion : then  half  the  log  of  this  remainder  subtracted  from  the 
log.  of  the  measured  distance  between  the  1st  and  2d,  or  the 
2d  and  3d  stations,  will  be  the  log.  of  the  height  of  the  object. 

PROBLEM  V. 


In  any  Spherical  Triangle,  knowing  Two  Sides  and  the  In- 
cluded Angle  ; it  is  required  to  find  the  Angle  Comprehend- 
ed by  the  Chords  of  those  two  sides. 

Let  the  angles  of  the  spherical  tri- 
angle be  a,  b,  c,  the  corresponding 
angles  included  by  the  chords  a',  b', 
c'  ; the  spherical  sides  opposite  the 
former  a,  6,  c,  the  chords  respect- 
ively opposite  the  latter  «,  /3,  y ; then, 
there  are  given  6,  c,  and  a,  to  find  a'. 

Here,  from  prob.  1 equa,  1 chap,  iv,  we  have 

cos  a = sin  6 sin  c cos  a + cos  b . cos  c. 

But  cos  c = cos  (ic  -f-  ic)  = cos2  |c  — sin2  Ac  (by  equa.  v 
ch  iii)  = (1— sin2  A,c) — sin2  ic  =1—2  sin2  Ac.  And  in  like 
manner  cos  a = 1 — 2 sin2  ic,  and  cos  6=1  — 2 sin2  i6. 
Therefore  the  preceding  equation  becomes 
1 — 2 sin2  la  = 4 sin  \b  . cos  16  . sin  |c  . cos  Ac 
(1  — 2 sin2  16)  (1  — 2 sin2  ic). 

But  sin  la  = i«,  sin  |6  = A/3,  sin  Ac  = iy  : which  values 
substituted  in  the  equation,  we  obtain,  after  a little  reduction, 
£2  + .>8  _*2 

/3y 


cos  A -j- 


2 X 


4 


Now, (equa.  ii  ch.  iii),  cos  a' 


cos 

& 


A 6 


cos  ic  , 

*j-  y J — a2 


2 


cos  a -f-  i/32  y2. 
. Therefore,  by 


substitution, 

/3y  . COS  a'  = /3y  . COS  $b  . COS  \c  . COS  A -f-  1/32  y2  ; 
whence,  dividing  by  )8y.  there  results 
cos  a'  = i 6 cos  ic  coS  a + ifl  . iy  ; 
or,  lastly,  by  restoring  the  values  of  A/3,  \y,  we  have 
cos  a'  = cos  A6  . cos  ic  . cos  a + sinA&  • sin  £c  . . , (1  ) 

Cor. 


73 


TRIGONOMETRICAL  SURVEYING. 


Cor  1 It  follows  evidently  from  this  formula,  that  when 
the  spherical  angle  is  right  or  obtuse,  it  is  always  greater  than 
the  corresponding  angle  of  the  chord? 

Cor.  2.  The  spherical  angle,  if  acute,  is  less  than  the  cor- 
responding angle  of  the  chords,  when  we  have  cos  a greater 
sin  y b . sin 


than 


1 — cos  J6 . sin  Jc 


PROBLEM  VL 


Knowing  Two  Sides  and  the  Included  Angle  of  a Rectilinear 
Triangle,  it  is  required  to  find  the  Spherical  Angle  of  the 
Two  Arcs  of  which  those  two  sides  are  the  chords. 

Here  j3,  y,  and  the  angle  a'  are  given,  to  find  a Now, 
since  in  all  cases,  cos  = (1  — sin2),  we  have 

cos  \b  . cos  ic  = y/  [(I  — sin2  ±b)  . (1  — sin2  J-c)]  ; 
we  have  also,  as  above,  sin  = i/3,  and  sin  ic  = ly. 
Substituting  these  values  in  the  equation  i of  the  preceding 
problem,  there  will  result,  by  reduction, 

COS  A — \&y 

COS  A — ^ (1_^)  . (1  + 46)T(T_:'>)  . (1  + lyY  • • ’ ' ) 

To  compute  by  this  formula,  the  values  of  the  sides  /3,  y, 
must  be  reduced  to  the  corresponding  values  ot  the  chords  of 
a circle  whose  radius  is  unity.  This  is  easily  effected  by  di- 
viding the  values  of  the  sides  given  in  feet,  or  toises,  & c by 
such  a power  of  10,  that  neither  of  the  sides  shall  exceed  2, 
the  value  of  the  greatest  chord,  when  radius  is  equal  to  unity. 

From  this  investigation,  and  that  of  the  preceding  problem, 
the  following  corollaries  may  be  drawn. 

Cor.  1.  If  c = b,  and  of  consequence  y = / 3,  then  will 
cos  a = cos  a cos2  |c  + sin2  4c  ; and  thence 
1 — 2 sin2  4a'  = (1—2  sin2  $a)  cos2  \c  - j-  ( 1 — cos2  _\c)  : 


from  which  may  be  reduced 

sin  4a'  = sin  \a  . cos  -i e . 
Cor  2.  Also,  since  cos  Lc  = ^/(l - 
equa.  ii  will,  in  this  case,  reduce  to 

sin  iA*: 

sin  4a  : 


..(III) 

.s1D2ac)  = v/0-^1 

• (IV).  . 


V3), 


✓(i  - i>)  • (i  + iy)  . 

Cor  3 From  the  equation  hi,  it  appears  that  the  vertical 
angle  of  an  isosceles  spherical  triangle,  is  always  greater  than 
the  corresponding  angle  ol  the  chords. 

Cor  4.  If  a = 90°,  the  formulas  i,  ii,  give 

cos  a = sin  sin  — 4/3y  . . . (V.) 

These  five  formulas  are  strict  and  rigorous,  whatever  be 
the  magnitude  of  the  triangle.  But  if  the  triangles  be  small, 
the  arcs  may  be  put  instead  of  the  sines  in  equa.  v,  then 

Cor  5.  As  cos  a' = sin  (90°  — a ) = in  this  case,  90°  — a' ; 
*he  small  excess  of  the  spherical  right  angle  over  the  corre- 
sponding 


GEODESIC  OPERATIONS. 


79 


sponding  rectilinear  angle,  will,  supposingffhe  arcs  b,  c,  taken 
in  seconds,  be  given  in  seconds  by  the-  following  expression 

90°  — a*  — — —rr (VI.) 

R 4R  V ' 

The  error  in  this  formula  will  not  amount  to  a second, 
when  b - f-  c is  less  than  10°,  or  than  700  miles  measured  on 
the  earth’s  surface. 

Cor.  6.  If  the  hypothenuse  does  not  exceed  11°,  we  may 
substitute  a sin  c instead  of  c,  and  a cos  c instead  of  b ; this 
will  give  be  = a2  . sin  c cos  c.  = \a2  . sin  2 (90°  — b)  =ia2 . 
sin  2b  ; whence 


(90“  _ A')  = 


<i2  . sin  2c  a2  . sin  2B 


• . (VII.) 


8r"  8r 

If  a and  b=c  = 46°  nearly  ; then  will  90°  — a'=  17".7. 

Cor  7 Retaining  the  same  hypothesis  of  a = 90°,  and 
# = or  < li°,  we  have 


b — b' 
Also  c — c'  = 


62  cot  B 

8r" 

be 

8r" 


be 

8r" 


. . . (VIII.) 
(LX.) 


Cor.  8.  Comparing  formulae  vm,  ix,  with  vi,  we  have  b — b' 
= c — c'  = a (90°  — a'.)  Whence  it  appears  that  the  sum  of 
the  two  excesses  of  the  oblique  spherical  angles,  over  the 
corresponding  angles  of  the  chords,  in  a small  right-angled  tri- 
angle, is  equal  to  the  excess  of  the  right  angle  over  the  cor- 
responding angle  of  the  chords.  So  that  either  of  the  formu- 
lae vr,  vii,  vm,  ix,  will  suffice  to  determine  the  difference  of 
each  of  the  three  angles  of  a small  right-angled  spherical  tri- 
angle, from  the  corresponding  angles  of  the  chords.  And 
hence  this  method  may  be  applied  to  the  measuring  an  arc  of 
the  meridian  by  means  of  a series  of  triangles.  See  arts.  8, 
9,  sect.  1 of  this  chapter. 


PROBLEM  VII. 


In  a Spherical  Triangle  abc,  Right  Angled  in  a,  knowing  the 
Hypothenuse  bc  (less  than  4°)  and  the  Angle  b,  it  is  required 
to  find  the  Error  e committed  through  finding  by  Plane 
Trigonometry,  the  Opposite  Side  ac. 

Referring  still  to  the  diagram  of  prob.  5,  where  we  now 
suppose  the  spherical  angle  a to  be  right,  we  have  (theor.  10 
chap  iv)  sin  b = sin  a . sin  b.  But  it  has  been  remarked  at 
pa.  381  vol.  i,  that  the  sine  of  any  arc  a is  equal  to  the  sum 
©f  the  following  series  ; 


AS 


a3  i 

Sin  A = A -f- 

2 o 2.o-4  o 


2345.67 

•r,sin*  = i“T  + W-^o+&c- 


&c. 


And, 


30 


TRIGONOMETRICAL  SURVEYING. 


And,  in  the  present  enquiry,  all  the  terms  after  the  second 
may  be  neglected,  because  the  6th  power  of  an  arc  of  4°  di- 
vided by  120,  gives  a quotient  not  exceeding  0"  . 01.  Con- 
sequently, we  may  assume  sin  6=6  — ^63,  sin  a = a — |a3: 
and  thus  the  preceding  equation  will  become, 

6 — i6 3 = sin  b (a  — ^a3) 
or  6 = a sin  b — i (a3  . sin  b — 63). 

Now,  if  the  triangle  were  considered  as  rectilinear,  we  should 
have  6 = a . sin  b;  a theorem  which  manifestly  gives  the 
side  6 or  ac  too  great  by  J-(a3  . sin  b — 63).  But,  neglecting 
quantities  of  the  fifth  order,  for  the  reason  already  assigned, 
the  last  equation,  but  one  gives  63  = a3  . sin3  b.  Therefore, 
by  substitution,  e = — £a3  . sin  b (1 — sin2  b)  : or,  to  have  this 
error  in  seconds,  take  r"  = the  radius  expressed  in  seconds, 

02  . CO' 2 B 

so  shall  e — — a . sin  b . — . 

6r  . R 

Cor.  1 If  a = 4°,  and  b = 35°  16',  in  which  case  the 
value  of  sin  b . cos2  b is  a maximum,  we  shall  find  e = - 4i'\ 
Cor.  2.  If,  with  the  same  data,  the  correction  be  applied, 
to  find  the  side  c adjacent  to  the  given  angle,  we  should  have 
, a2  , sinz  e 

e = a . cos  b — - — — — . 

3r  r 

So  that  this  error  exists  in  a contrary  sense  to  the  other  ; the 
one  being  subtractive,  the  other  additive. 

Cor.  3.  The  data  being  the  same,  if  we  have  to  find  the 
angle  c,  the  error  to  be  corrected  will  be 
/,  „ si'1  2 b 

e = a2  . - 

4r 

As  to  the  excess  of  the  arc  over  its  chord,  it  is  easy  to  find  it 
correctly  from  the  expressions  in  prob.  5 : but  for  arcs  that 
are  very  small,  compared  with  the  radius,  a near  approxima- 
tion to  that  excess  will  be  found  in  the  same  measures  as  the 
radius  of  the  earth,  by  taking  of  the  quotient  of  the  cube 
of  the  length  of  the  arc  divided  by  the  square  of  the  radius. 

PROBLEM  vnr. 

It  is  required  to  Investigate  a Theorem,  by  means  of  which. 
Spherical  Triangles,  whose  Sides  are  Small  compared  with 
the  radius,  may  be  solved  by  the  rules  for  Plane  Trigono- 
metry, without  considering  the  Chords  of  the  respective 
Arcs  or  Sides. 

Let  a,  6,  c,  be  the  sides,  and  a,  b,  c,  the  angles  of  a sphe- 
rical triangle,  on  the  surface  of  a sphere  whose  radius  is  r 

then 


SPHERICAL  EXCESS. 


81 


then  a similar  triangle  on  the  surface  of  a sphere  whose  radius 

= 1,  will  have  for  its  sides  -,  -,  - ; which,  for  the  sake 

r r r 

of  brevity,  we  represent  by  «,  y,  respectively  : then,  by 

. Cus  a — COS  . COS'  y 

equa.  i,  chap,  iv,  we  have  cos  a = — T . 

r »m  /s  . sin  y 

Now,  r being  very  great  with  respect  to  the  sides,  a,  b,  c, 

we  may,  as  in  the  investigation  of  the  last  problem,  omit  all 

the  terms  containing  higher  than  4th  powers,  in  the  series 

for  the  sine  and  cosine  of  an  arc,  given  at  pa.  381  vol.  i : so 

shall  we  have,  without  perceptible  error, 

a2  , st  4 . 

cos  ^ = 1 . . . sin  /3  = & — — . 

2 2.3.4  2.3 

And  similar  expressions  may  be  adopted  for  cos  /3,  cos  y, 
sin  y.  Thus,  the  preceding  equation  will  become 

_ _ 10*2  +>2-*2  ) + i?  (*4-04 -y*  )-P-  y2 

-}y2 

Multiplying  both  terms  ot  this  fraction  by  l+£  (/32  + y2),  to 
simplify  the  denominator,  and  reducing,  there  will  result, 

@2  Jf- yi  — a2  f et4-J-&4  “f-^4 — yS2  — 2ct 2 y-  — tsQ^y2 
GOS  A _ , [ ’ ~T~n  ' • 

2 fay  24$ y 

Here,  restoring  the  values  of  «,  (i , y,  the  second  member  of 
the  equation  will  be  entirely  constituted  of  like  combinations 
©f  the  letters,  and  therefore  the  whole  may  be  represented  by 

C0S  A “ Wc+M* ^ 

Let,  now,  a'  represent  the  angle  opposite  to  the  side  a,  in 
the  rectilinear  triangle  whose  sides  are  equal  in  length  to  the 
arcs  a,  b,  c ; and  we  shall  have 

. b 2 +c2  — ai  m 

C0S  A = — "2Tc =2bcf 

Squaring  this,  and  substituting  for  cos2  a its  value  1 — sin2  a', 
there  will  result 

— 462  c-  sin2  a '=  a2 -f-62 -f-c2 — 2a2  62—  2a2  c2—  2b2  c2=  n. 
So  that,  equa.  x,  reduces  to  the  form 

» be  . , , 

cos  a — cos  a — — sin2  a . 

6>  2 

Let  a = a'  -f-  Jf,  then,  as  x is  necessarily  very  small,  its  second 
power  may  be  rejected,  and  we  may  assume  cos  a = cos  a — 
x . sin  a'  : whence,  substituting  for  cos  a this  value  of  it,  we 

shall  have  x = . sin  a. 

6l’2 

It  hence  appears  that  x is  of  the  second  order,  with  respect 

b c 

to  — and  — ; and  of  course  that  the  result  is  exact  to  quan- 


Vol.  II. 


12 


titiea 


TRIGONOMETRICAL  SURVEYING. 


82 


Therefore,  because  a=a'+x. 


tides  within  the  fourth  order. 

, . be 
A = A + — . Sin  A . 

6r2 

But,  by  prob.  2 rule  2,  Mensuration  of  Planes  i&c  sin  a'  is 
the  area  of  the  rectilinear  triangle,  whose  sides  are  o,  b,  and  c 

Therefore  a — a -f-  ; 

or2 

area 

or  a'  = a — — . 

3 /■* 

In  like 


manner 


> , area 

ike  5 B = B - *3. 

‘ner  ( area 

' c'  = c — — — . 


And  a1  + b'  + c'  = 180c 


, , area 

= A + B + C- — . 

r1 


or,  ~ = A + B + c — 180°. 

Whence,  since  the  spherical  excess  is  a measure  of  the  area 
(th.  5 ch.  iv),  we  have  this  theorem  : viz. 

A spherical  triangle  being  proposed , of  which  the  sides  are 
very  small , compared  with  the  radius  of  the  sphere  ; if  from 
each  of  its  angles  one  third  of  the  excess  of  the  sum  of  its 
three  angles  above  two  right  angles  be  subtracted , the  angles 
so  diminished  may  be  taken  for  the  angles  of  a rectilineal 
triangle,  whose  sides  are  equal  in  length  to  those  of  the  pro- 
posed spherical  triangle*. 

Scholium. 

We  have  already  given,  at  th  5 chap,  iv,  expressions  foi 
finding  the  spherical  excess,  in  the  two  cases,  where  two  sides 
and  the  included  angle  of  a triangle  are  known,  and  where 
the  three  sides  are  known.  A few  additional  rules  may  with 
propriety  be  presented  here. 

1.  The  spherical  excess  e,  may  be  found  in  seconds,  by  the 

expression  e = ; where  s is  the  surface  of  the  triangle  = 

.7  • , , . . T „ sin  b sin  c 

\bc  . sin  a = bab  . sin  c = . sin  b = ba2. ; — - — - , r is 

2 - 2 sin  (b  + c) 

the  radius  of  the  earth,  in  the  same  measures  as  a,  b , and  c, 
and  r"  = 206264"-8,  the  seconds  in  an  arc  equal  in  length  to 
the  radius. 

If  this  formula  be  applied  logarithmically  ; then  log.  r"  = 
log.  __i_  = 5-3144251. 


* This  curious  theorem  was  first  announced  by  M.  Legendre,  in  the 
Memoirs  of  the  Paris  Academy,  for  1787.  Legendre’s  investigation 
•is  neatly  the  same  as  the  above  : a shorter  investigation  is  given  by 
Swanberg,  at  pa.  40,  of  his  “ Esposiii  n des  Operations  faitesen  Lap- 
ponie  but  it  is  defective  in  point  of  perspicuity. 

2.  From 


SPHERICAL  EXCESS. 


nn 

OO 


2.  From  the  logarithm  of  the  area  of  the  triangle,  taken 
as  a plane  one,  in  feet,  subtract  the  constant  log  9 3267737 
then  the  remainder  is  the  logarithm  of  the  excess  above  180° 
in  seconds  nearly*. 

3.  Since  s = \ be  . sin  a,  we  shall  manifestly  have  e = 


11  L 

■—  be  . sin  a. 

2r2 


Hence,  if  from  the  vertical  angle  b we  demit 


the  perpendicular  Bn  upon  the  base  ac,  dividing  it  into  the 
two  segments.*,  /3,  we  shall  have  b = «+/3,  jj 


and  thence  e - c(*  -f-  /3)  sin  a = * „ *c. 

sin  a -f-  /3c  . sin  a.  But  the  two  right- 

angled  triangles  abd,  cbd,  being  nearly  rec- 
tilinear, give  * = a . cos  c,  and  /3  =*=  c . cos  a; 
whence  we  have 


e = — ac 

2r2 


+ X\  0 

c- 

2r2 


. Sin  A . COS  A. 


in  like  manner,  the  triangle  abc,  which  itself  is  so  small  as  to 
differ  but  little  from  a plane  triangle,  gives  c . sin  a = a . sin  c. 
Also,  sin  a . cos  a = | sin  2a,  and  sin  c . cos  c = i sin  2c 
(equa.  xv.  ch.  iii).  Therefore,  finally, 

e = — a2  . sin  2c  4-  — c2  . sin  2a. 

4r2  4 7-s 


Frbm  this  theorem  a table  may  be  formed,  from  which  the 
spherical  excess  may  be  found  ; entering  the  table  with  each 
of  the  sides  above  the  base  and  its  adjacent  angle,  as  argu- 
ments. 

4.  If  the  base  b and  height  h,  of  the  triangle  are  given, 
then  we  have  evidently  e = ±bh  . Hence  results  the  fol- 


lowing simple  logarithmic  rule  : Add  the  logarithm  of  the 
base  of  the  triangle,  taken  in  feet,  to  the  logarithm  of  the 
perpendicular,  taken  in  the  same  measure  deduct  from  the 
sum  the  logarithm  9,6278037  ; the  remainder  will  be  the 
common  logarithm  of  the  spherical  excess  in  seconds  and 
decimals. 

5.  Lastly,  when  the  three  sides  of  the  triangle  are  given 
in  feet  : add  to  the  logarithm  of  half  their  sum,  the  logs,  of 
the  three  differences  of  those  sides  and  that  half  sum,  divide 
the  total  of  these  4 logs,  by  2,  and  from  the  quotient  subtract 
the  log.  9 3267737  ; the  remainder  will  be  the  logarithm  of 
the  spherical  excess  in  seconds  &c.  as  before. 

One  or  other  of  these  rules  will  apply  to  all  cases  in  which 
the  spherical  excess  will  be  required. 


* This  is  General  Hoy’s  rule  given  in  the  Philosophical  nsac- 
tionsj  for  1790,  pa.  171.  PROBLEM 


34 


TRIGONOMETRICAL  SURVEYING. 


PROBLEM  IX. 

Given  the  Measure  of  a Base  on  any  Elevated  Level  ; to  find 
its  Measure  when  Reduced  to  the  Level  of  the  Sea. 

Let  r represent  the  radius  of  the  earth,  or  the  distance 
from  its  centre  to  the  surface  of  the  sea,  r + h the  radius  re- 
ferred to  the  level  of  the  base  measured,  the  altitude  h being 
determined  by  the  rule  for  the  measurement  of  such  altitudes 
by  the  barometer  and  thermometer,  (in  this  volume)  ; letB  be 
the  length  of  the  base  measured  at  the  elevation  hs  and  b that 
of  the  base  referred  to  the  level  of  the  sea. 

Then  because  the  measured  base  is  all  along 
reduced  to  the  horizontal  plane,  the  two,  b 
and  b,  will  be  concentric  and  similar  arcs,  to  the 
respective  radii  r + h and  r.  Therefore,  since 
similar  arcs,  w'hether  of  spheres  or  spheriods, 
are  as  their  radii  of  curvature,  we  have 

r -{-  h : r : : b : b = — 

>•+  « 

Hence,  also  b — b = b — = : or,  by  actually  divi- 

r- t~A  r-{-  h ■ 

ding  b h by  r + h,  we  shall  have 

li  h 2 hs  h* 

B—b  = b X ( 1 r &c.) 

r r2  r3  ri 

Which  is  an  accurate  expression  for  the  excess  of  b above  b. 

But  the  mean  radius  of  the  earth  being  more  than  21  mil- 
lion feet,  if  h the  difference  of  level  were  50  feet,  the  second 
and  all  succeeding  terms  of  the  series  could  never  exceed 
the  fraction  rtWo  rrVo^o  o o 1 an^  may  therefore  safely  be  neg- 
lected : so  that  for  all  practical  purposes  we  may  assume 
n h • 

B—b  — — . Or,  in  logarithms,  add  the  logarithm  of  the 

measured  base  in  feet,  to  the  logarithm  of  its  height  above 
the  level  of  the  sea,  subtract  from  the  sum  the  logarithm 
7-3223947,  the  remainder  will  be  the  logarithm  of  a number, 
which  taken  from  the  measured  base  will  leave  the  reduced 
base  required. 

PROBLEM  X. 

To  determine  the  Horizontal  Refraction. 

1.  Particles  of  light,  in  passing  from  any  object  through 
the  atmosphere,  or  part  of  it.  to  the  eye,  do  not  proceed  in  a 
right  line  ; but  the  atmosphere  being  composed  of  an  infini- 
tude of  strata  (if  we  may'  so  call  them)  whose  density  increases 
as  they  are  posited  nearer  the  earth,  the  luminous  rays  which 

pass 


GEODESIC  OPERATIONS. 


85 


pass  through  it  are  acted  on  as  if  they  passed  successively 
through  media  of  increasing  density,  and  are  therefore  inflect- 
ed more  and  more  towards  the  earth  as  the  density  augments. 
In  consequence  of  this  it  is.  that  rays  from  objects,  whether 
celestial  or  terrestrial,  proceed  in  curves  which  are  concave  to- 
wards the  earth  ; and  thus  it  happens,  since  the  eye  always  re- 
fers the  place  of  objects  to  the  direction  in  which  the  rays 
reach  the  eye,  that  is,  to  the  direction  of  the  tangent  to  the 
curve  at  that  point,  that  the  apparent,  or  observed  elevations 
of  objects,  are  always  greater  than  the  true  ones.  The  differ- 
ence of  these  elevations,  which  is,  in  fact,  the  effect  of  refrac- 
tion, is,  for  the  sake  of  brevity,  called  refraction  : and  it  is  dis- 
tinguished into  two  kinds,  horizontal  or  terrestrial  refraction, 
being  that  which  affects  the  altitudes  of  hills,  towers,  and 
other  objects  on  the  earth’s  surface  ; and  astronomical  refrac- 
tion, or  that  which  is  observed  with  regard  to  the  altitudes  of 
the  heavenly  bodies.  Refraction  is  found  to  vary  with  the  state 
of  the  atmosphere,  in  regard  to  heat  or  cold,  humidity  or  dry- 
ness, &c  : so  that,  determinations  obtained  for  one  state  of  the 
atmosphere,  will  not  answer  correctly 'for  another,  without 
modification  Tables  commonly  exhibit  the  refraction  at  dif- 
ferent altitudes,  for  some  assumed  mean  state. 

2.  With  regard  to  the  horizontal  refraction  the  following 
method  of  determining  it  has  been  successfully  practised  in 
the  English  Trigonometrical  Survey. 

Let  a,  a',  be  two  elevated  stations  on 
the  surface  of  the  earth,  bd  the  inter- 
cepted arc  of  the  earth’s  surface,  c the 
earth’s  centre,  ah',  a'h,  the  horizontal 
lines  at  a.  a',  produced  to  meet  the  oppo- 
site vertical  lines  cn',  ch.  Let  a,  a',  re- 
presentthe  apparent  places  of  the  objects 
a,  a',  then  is  o'aa  the  refraction  observ- 
ed at  a,  and  ax' a the  refraction  observed  at  a'  ; and  half  the 
sum  of  those  angles  will  be  the  horizontal  refraction,  if  we  as- 
sume it  equal  at  each  station. 

Now,  an  instrument  being  placed  at  each  of  the  stations 
a,  a',  the  reciprocal  observations  are  made  at  the  same  in- 
stant of  time,  which  is  determined  by  means  of  signals  or 
watches  previously  regulated  for  that  purpose  ; that  is,  the 
observer  at  a takes  the  apparent  depression  of  a',  at  the  same 
moment  that  the  other  observer  takes  the  apparent  depression 
of  A. 

In  the  quadrilateral  aca'i,  the  two  angles  a,  a',  are  right 
angles,  an  1 therefore  the  angles  : a id  c are  together  equal  to 
two  right  angles  : but  the  three  angles  of  the  triangle  iaa' 

are 


96 


TRIGONOMETRICAL  SURVEYING. 


are  together  equal  to  two  right  angles  ; and  consequently  the 
angles  a aDd  a'  are  together  equal  to  the  angle  c,  which  is 
measured  by  the  arc  bd.  If  therefore  the  sum  of  the  two 
depressions  ha  'a,  h'a  a',  be  taken  from  the  sum  of  the  angles 
ha'ah'aa'  or,  which  is  equivalent,  from  the  angle  c (which, 
is  known,  because  its  measure  bd  is  known)  ; the  remainder 
is  the  sum  of  both  refractions,  or  angles  oa'a,  a' aa  . Hence 
this  rule,  take  the  sum  of  the  two  depressions  from  the  measure 
of  the  intercepted  terrestrial  arc , half  the  remainder  is  the  re- 
jraction.  1 

3.  If,  b}'  reason  of  the  minuteness  of  the  contained  arc  bd, 
one  of  the  objects,  instead  of  being  depressed,  appears  ele- 
vated, as  suppose  a to  a"  : then  the  sum  of  the  angles  a"  aa 
and  a a' a will  be  greater  than  the  sum  iaa'+ia'a.  or  than  c, 
by  the  angle  of  elevation  o"  aa'  ; but  if  from  the  former  sum 
there  be  taken  the  depression  ha'a,  there  will  remain  the 
sum  of  the  two  refractions.  So  that  in  this  case  the  rule  be- 
comes as  follows  : take  the  depression  from  the  sum  of  the  con- 
tained arc  and  elevation,  half  the  remainder  is  the  refraction. 

5.  The  quantity  of  this  terrestrial  refraction  is  estimated 
by  Dr.  Maskelyne  at  one-tenth  of  the  distance  of  the  object 
observed  expressed  in  degrees  of  a great  circle  So,  if  the 
distance  be  10000  fathoms,  its.  10th  part  1000  fathoms,  is  the 
60  part  of  a degree  of  a great  circle  on  the  earth,  or  1',  which 
therefore  is  the  refraction  in  the  altitude  of  the  object  at  that 
distance. 

But  M.  Legendre -is  induced,  he  says,  by  several  experi- 
ments, to  allow  only  T'?th  part  of  the  distance  for  the  refrac- 
tion in  altitude.  So  that,  on  the  distance  of  10000  fathoms, 
the  14th  part  of  which  is  714  fathoms,  he  allows  only  44"  of 
terrestrial  refraction,  so  many  being  contained  in  the  714 
fathoms.  See  his  Memoir  concerning  the  Trigonometrical 
operations,  &c. 

Again,  M.  Delambre,  an  ingenious  French  astronomer, 
makes  the  quantity  of  the  terrestrial  refraction  to  be  the  1 Ith 
part  of  the  arch  of  distance.  But  the  English  measurers, 
especially  Col.  Mudge,  from  a multitude  of  exact  observations, 
determine  the  quantity  of  the  medium  refraction,  to  be  the 
12th  part  of  the  said  distance. 

The  quantity  of  this  refraction,  however,  is  found  to  vary 
considerably,  with  the  different  states  of  the  weather  and  at- 
mosphere, from  the  |th  to  the  T’Tth  of  the. contained  arc.  See 
Trigonometrical  Survey,  vol.  1 pa.  160,  365. 


Scholium 


REFRACTION 


87 


Scholium. 


Having  given  the  mean  results  of  observations  on  the  ter- 
restrial refraction,  it  may  not  be  amiss,  though  we  cannot 
enter  at  large  into  the  investigation,  to  present  here  a correct 
table  of  mean  astronomical  refractions.  The  table  which  has 
been  most  commonly  given  in  books  of  astronomy  is  Dr. 
Bradley’s,  computed  from  the  rule  r = 57"  X cot  (a  + 3 r), 
where  a is  the  altitude,  r the  refraction,  and  r = 2'  35"  when 
a = 20°.  But  it  has  been  found  by  numerous  observations, 
that  the  refractions  thus  computed  are  rather  too  small. — 
Laplace,  in  his  Mecanique  Celeste  (tome  iv  pa.  27)  deduces 
a formula  which  is  strictly  similar  to  Bradley’s  ; for  it  is 
r = m X tan  (2  — nr),  where  2 is  the  zenith  distance,  and  in 
and  n are  two  constant  quantities  to  be  determined  from  ob- 
servation The  only  advantage  of  the  formula  given  by  the 
French  philosopher,  over  that  given  by  the  English  astrono- 
mer, is  that  Laplace  and  his  colleagues  have  found  more 
correct  coefficients  than  Bradley  had. 

Now,  if  r = 57°-2957795,  the  arc  equal  to  the  radius,  if 

we  make  m = — , (where  k is  a constant  coefficient  which,  as 

n 

well  as  n,  is  an  abstract  number,)  the  preceding  equation  will 
become  = k X tan  (2— nr).  Here,  as  the  refraction  r is 
always  very  small,  as  well  as  the  correction  nr,  the  trigono- 
metrical tangent  of  the  arc  nr  may  be  substituted  for  — ; thus 

we  shall  have  tan  nr  = k . tan  (2— nr). 

But  nr  = \z  — (^2 — nr)  ....  2 — nr  — \ 2 -{-  (^2— nr)  ; 


Conseq. 


tan  nr 
tan  (2—  nr) 


, ,S  z — 2nr 

tan  ( . 

^2  2 sin  -z  -sin  (z  — 2nr)  j 


tan  2 


2 , 2 — 2 nr  sin  2 + 3111(2  — 2nr) 
2 


\-k  . 

Hence,  sin  (2 — 2 nr)  — sin  2. 

j & 

This  formula  is  easy  to  use,  when  the  co-efficients  n and 


are  known  : and  it  has  been  ascertained,  by  a mean  of  many 
observations,  that  these  are  4 and  -99765175  respectively. 
Thus  Laplace’s  equation  becomes 

sin  (z_8r)  = -99765175  sin  2 : 
and  from  this  the  following  table  has  been  computed.  Besides 
the  refractions,  the  differences  of  refraction,  for  every  10 
minutes  of  altitude,  are  given  ; an  addition  which  will  render 
the  table  more  extensively  useful  in  all  cases  where  great 
accuracy  is  required.  - 


88 


TRIGONOMETRICAL  SURVEYING 


Table  of  Refractions. 

Barom.  29  92  inc.  Fah.  Thermom.  54°. 


Alt 

app- 

Refrac 

, Diff 
■ on  10 

Alt 

app- 

I Reft-. 

Diff 

,10. 

Alt. 

app 

Retr. 

Diff; 

10’. 

Alt. 

! app. 

Ref 

D ft 
108 

E 

. M. 

M 

s. 

s. 

D.  31. 

M.  S. 

' S. 

| D. 

M 

. s 

S. 

D 

s. 

0-25 

0 

0 

33 

46-3 

112-0 

7 0 

7 24-8 

9-5 

14 

3 

49-8 

2-58 

56 

39-3 

0-24 

10 

31 

54*3 

105-0 

10 

7 15-3 

9-0 

15 

3 

34-3 

2-28 

57 

37-8 

0-24 

20 

30 

9o 

1 97-3 

20 

7 6-3 

8-6 

16 

3 

20-6 

202 

58 

36-4 

0-23 

30 

28 

32-1 

i 89-8 

30 

6 57-7 

8-1 

17 

■3 

8-5 

1-82 

59 

350 

0-22 

40 

2 7 

2-2 

! 83-6 

40 

6 49-6 

Tv 

18 

2 

57-6 

1-65 

60 

35-6 

0 2-2 

50 

25 

■8-6 

77-4 

■ 50 

6 41-9 

7-5 

19 

2 

47  7 

1-48 

61 

32-3 

0-21 

i 

0 

24 

-1'2 

7:-6 

8 0 

6 34-4  l 7-3 

20 

2 

38-8 

1-37 

62 

310 

0-21 

10 

23 

9.6 

| 66-2 

1016  27-1 

7-1 

21 

2 

30-i' 

1-24 

63 

29-7 

0-20 

20 

22 

34 

j 61-5 

■ 20 

| 6 2r'-0 

j 6-9 

22 

2 

23- 

Ml 

64 

284 

0-20 

30 

21 

1-9 

1 57-1 

; 30 

6 13-1 

; 6-7 

23 

2 

6-5 

105 

65 

27-2 

O-20 

40 

20 

4-8 

| 53-3 

40 

1 6 64 

6-5 

24 

2 

10-2 

0-98 

66 

25-9 

0-20 

50 

19 

1-5 

I 49-3 

50 

l 5 59*9 

6-3 

25 

2 

4-3 

0-90 

67 

24-7 

0-20 

2 

0 

18 

22-2 

1 45-9 

9 0 

5 53*6 

6-2 

26 

58-9 

0-83 

68 

23-5 

0-20 

10 

17 

36-3 

43-1 

10 

5 47*4 

5-9 

27 

1 

53-9 

0-78 

69 

22-4 

0-20 

20 

16 

53:2 

I 39-8 

; 20 

5 41-5 

! 5-7 

28 

1 

49-2 

0-73 

70 

21  2 

01 9 

30 

16 

13-4 

37-4 

j 30 

5 35-8 

5-5 

29 

1 

44-8 

0-70 

7 

20-0 

018 

40 

15 

36-0 

35-1 

! 40 

5 30-3 

i 5-3 

30 

1 

4l'-6 

0-65 

72 

1 8-9 

0-8 

50 

15 

0-9 

32-8 

I 50  i 5 25-0 

5*2  1 

31 

I 

3G  7 

0-60 

73 

17-8 

0-18 

3 

0 

14 

28-1 

30-8 

10  0 i 5 19-8 

51  . 

32 

1 

33-1 

0-58 

74 

16-7 

0-18 

10 

13 

57-3 

28-8 

1 10  ! 5 14-7 

5-0 

33 

1 

29-6 

0-56 

75 

15-6 

01 7 

20 

13 

28-5 

27-2 

1 20 

5 9-7 

4-8  1 

34 

1 

26-2 

0-53 

76 

14-5 

0-17 

30 

13 

1-3 

25-7 

1 30 

5 4-9 

46  ! 

35 

1 

231 

0-50 

77 

13*5 

0-17 

40 

12 

35-6 

24-3! 

40 

5 0-3 

4-4 

36 

1 

20-1 

048 

1 78 

12  4 

0-17 

50 

12 

11-3 

23-0 

50 

4 55*9 

4-2  i 

37 

1 

17-. 

0-47 

! 79 

1 1-3 

0-17 

4 

0 

11 

48-3 

21-7| 

11  0 

4 51-7 

4 1 1 

38 

1 

144 

043 

SO 

10-5 

0-17 

10 

11 

26-6 

20:5 

10 

4 47-6 

4-0 

39 

1 

11  8 

0-42 

' 81 

9*5 

0-17 

20 

11 

61 

19-4 

20 

4 43-6 

4-0  ! 

40 

1 

9-3 

U-40 

82 

8-2 

0-17 

30 

10 

46-7 

18-4  1 30 

4 39-6 

3-9  1 

41 

1 

6-9 

0-38 

S3 

7-2 

0-17 

40 

10 

28-3 

17-4 

40 

4 35*7 

3-9  1 

42 

1 

4-6 

0-37 

84 

6-1 

0-17 

50 

10 

10-9 

1 6-6 

50 

4 31  8 

3-8  ' 

43 

1 

2-4 

035 

85 

5-1  * 0-17 

5 

0 

9 

54-3 

1 5*9 

12  0 

4 28-0 

3-7  | 

44 

1 

0-3 

0-34 

66 

4-1 

0-17 

10 

9 

38-4 

15-0 

10 

4 24-3 

3-6 

45 

0 

58-2 

0*33 

87 

3-1  j 0-17 

20 

9 

23-4 

144 

20 

4 20’ 7 

3*5  ; 

46 

0 

56  2 

0-32 

88 

20 1 0-17 

30 

9 

9-0 

13-7 

30 

4 17-2 

3*4 

47 

0 

54  3 

0-31 

89 

10 

0-17 

40 

8 

55*3 

13-0 

40! 

4 13-8 

3-2 

4$ 

0 

52-4 

0-30 

90 

OO  1 

50 

8 

42-3 

124 

50 

4 10-6 

31 

49’ 

0 

50-6 

0-29 

6 

0 

8 

29-9 

1I-S 

13  0 

4 7*5 

3*1 

50 

0 

4S-9 

028 

10 

8 

18-1 

11  5 

10 

4 4-4 

S-0  1 

51 

0 

47-2 

0-27 

20 

8 

6-6 

11*0 

20 

4 1-4 

30 

52 

0 

45-5 

0-26 

30 

7 

55-6 

1 0-6  j 

30 

3 58-4 

2-9  ! 

53 

0 

43-9 

0-26 

40 

7 

45 '0 

1-1-3 

40 

3 55-5 

2-9 

54 

0 

42  3 

0-25 

50 

7 

34-7 

9-9  1 

50 

3 52-6 

2-8 

55 

0 

40-8 

0-25 

7 

0 

7 

24-8 

li 

14  0 

3 49-8 

* 

56 

0 

39-3 

PROBLEM 


REFRACTION. 


80 


PROBLEM  XI. 

To  find  the  Angle  made  by  a Given  Line  with  the  Meridian. 

1.  The  easiest  method  of  finding  the  angular  distance  of  a 
given  line  from  the  meridian,  is  to  measure  the  greatest  and 
the  least  angular  distance  of  the  vertical  plane  in  which  is  the 
star  marked  a in  Ursa  minor  (commonly  called  the  pole  star ), 
from  the  said  line  : for  half  the  sum  of  these  two  measures 
will  manifestly  be  the  angle  required. 

2.  Another  method  is  to  observe  when  the  sun  is  on  the 
given  line  ; to  measure  the  altitude  of  his  centre  at  that  time, 
and  correct  it  for  refraction  and  parallax.  Then,  in  the  sphe- 
rical triangle  zps,  where  z is  the  zenith 
of  the  place  of  observation,  p the  ele- 
vated pole,  and  s the  centre  of  the 
sun,  there  are  supposed  given  zs  the 
zenith  distance,  or  co-altitude  of  the 
sun,  ps  the  co-declination  of  that  lu- 
minary, pz  the  co-latitude  of  the  place  of  observation,  and 
zps  the  hour  angle,  measured  at  the  rate  of  15°  to  an  hour, 
to  find  the  angle  szp  between  the  meridian  pz  and  the  ver- 
tical zs,  on  which  the  sun  is  at  the  given  time.  And  here, 
as  three  sides  and  one  angle  are  known,  the  required  angle  is 
readily  found,  by  saying,  as  sine  zs  : sine  zps  : : sine  ps  ; 
sine  pzs  ; that  is,  as  the  cosine  of  the  sun’s  altitude,  is  to  the 
sine  of  the  hour  angle  from  noon  ; so  is  the  cosine  of  the  sun’s 
declination,  to  the  sine  of  the  angle  made  by  the  given  verti- 
cal and  the  meridian. 

Note.  Many  other  methods  are  given  in  books  of  Astrono- 
my ; but  the  above  are  sufficient  for  our  present  purpose. 
The  first  is  independent  of  the  latitude  of  the  place  ; the  se- 
cond requires  it. 

PROBLEM  Xn. 

To  find  the  latitude  of  a Place. 

The  latitude  of  a place  may  be  found  by  observing  the 
greatest  and  least  altitude  of  a circumpolar  star,  and  then  ap- 
plying to  each  the  correction  for  refraction  : so  shall  half  the 
sum  of  the  altitudes,  thus  corrected,  be  the  altitude  of,  the 
pole,  or  the  latitude, 

Vor..  II. 


13 


For, 


90 


TRIGONOMETRICAL  SURVEYING. 


For,  if  p be  the  elevated  pole,  st 
the  circle  described  by  the  star,  fr 
= ez  the  latitude  : then  since  ps  = 
p<,  pr  must  be  = i (r£-|-rs). 

This  method  is  obviously  indepen- 
dent of  the  declination  of  the  star  : 
it  is  therefore  most  commonly  adopt- 
ed in  trigonometrical  surveys,  in 
which  the  telescopes  employed  are 
of  such  power  as  to  enable  the  observer  to  see  stars  in  the  day- 
time : the  pole-star  being  here  also  made  use  of. 

Numerous  other  methods  of  solving  this  problem  likewise 
are  given  in  books  of  Astronomy  ; but  they  need  not  be  de- 
tailed here. 

Carol.  If  the  mean  altitude  of  a circumpolar  star  be  thus 
measured,  at  the  two  extremities  of  any  arc  of  a meridian,  the 
difference  of  the  altitudes  will  be  the  measure  of  that  arc  : 
and  if  it  be  a small  arc,  one  for  example  not  exceeding  a de- 
gree of  the  terrestrial  meridian,  since  such  small  arcs  differ 
extremely  little  from  arcs  of  the  circle  of  curvature  at  their 
middle  points,  we  may,  by  a simple  proportion,  infer  the 
length  of  a degree  whose  middle  point  is  the  middle  of  that 
arc. 

Seholium. 

Though  it  is  not  consistent  with  the  purpose  of  this  chap- 
ter to  enter  largely  into  the  doctrine  of  astronomical  spherical 
problems  ; yet  it  may  be  here  added,  for  the  sake  of  the  young 
student  that  if  a = rfght  ascension,  d = declination,  l = 
latitude,  A = longitude,  p = angle  of  position  (or,  the  angle 
at  a heavenly  body  formed  by  two  great  circles,  one  passing 
through  the  pole  of  the  equator  and  the  other  through  the 
pole  of  the  ecliptic),*  = inclination  or  obliquity  of  the  eclip- 
tic, then  the  following  equations,  most  of  which  are  new,  ob- 
tain generally,  for  all  the  stars  and  heavenly  bodies. 

1.  tan  a = tan  A . cos  i— tan  l . sec  A . sin  i. 

2.  sin  d — sin  A . cos  l . sin  f+sin  l . cos  i. 

3.  tan  A = sin  i . tan  d . sec  o+tan  a . cos  i. 

4.  sin  l = sin  d : cos  i — sin  a . cos  d . sin  i. 

5.  cotan  p — cos  d . sec  a . cot  i+sin  d . tan  a. 

6.  cotan  p = cos  l . sec  A . cot  i — sin  / . tan  A. 

7.  cos  a . cos  d = cos  l . cos  A. 

8.  sin  p . cos  d = sin  i . cos  A. 

9.  sin  p . cos  A = sin  i . cos  a. 

10.  tan  a = tan  A . cos  i.  ) when  Z = 0,  as  is  always  the  case 

11.  cos  A = cos  a . cos  d.  ( with  the  sun. 


The 


TO  FIND  THE  LATITUDE. 


91 


The  investigation  of  these  equations,  which  is  omitted  for 
iihe  sake  of  brevity,  depends  on  the  resolution  of  the  spheri- 
cal triangle  whose  angles  are  at  the  poles  of  the  ecliptic  and 
equator,  and  the  given  star,  or  luminary. 

PROBLEM  Xin, 


To  determine  the  Ratio  of  the  Earth’s  Axes,  and  their  Actual 
Magnitude,  from  the  Measure  of  a Degree  or  Smaller 
Portion  of  a Meridian  in  Two  Given  Latitudes  ; the  earth 
being  supposed  a spheroid  generated  by  the  rotation  of  an 
ellipse  upon  its  minor  axis. 

Let  adbe  represent  a meridian 
of  the  earth,  de  its  minor  axis, 
ab  a diameter  of  the  equator, 
m,  m,  arcs  of  the  same  number 
of  degrees,  or  the  same  parts  of 
a degree,  of  which  the  lengths 
are  measured,  and  which  are  so 
Small,  compared  with  the  mag- 
nitude of  the  earth,  that  they 
may  be  considered  as  coinciding  with  arcs  of  the  osculatory 
circles  at  their  respective  middle  points  ; let  mo,  mo,  the  radii 
of  curvature  of  those  middle  points,  be=R  and  r respectively  ; 
mp,  mp,  ordinates  perpendicular  to  ab  : suppose  further  cd=c, 
cb  = d ; d2  — c2  = e2  ce  = x ; c p = u ; the  radius  or  sine 
total  = ] ; the  known  angle  bsm,  or  the  latitude  of  the  mid- 
dle point  m,  = l ; the  known  angle  b s/n,  or  the  latitude  of  the 
point  m = / ; the  measured  lengths  of  the  arcs  m and  m be- 
ing denoted  by  those  letters  respectively. 

Now  the  similar  sectors  whose  arcs  are  m,  m,  and  radii  of 
curvature  r,  r,  give  r : r : : m : m ; and  consequently  r m = 
tm.  The  central  equation  to  the  ellipse  investigated  at  p.  533 

of  volume  first  gives  pm  = c-y/  (d2  — x2)-,  pm—  ^^/(cZ2-m2); 

0 ^ ^ C 2 2/ 

also  sp  = — sp  =— ■-  (by  th.  17  Ellipse).  And  the  method 


of  finding  the  radius  of  curvature  (Flux.  art.  74,  75),  ap- 
plied to  the  central  equations  above,  gives 


(rf4  — e2  2 


; and  r = 


(^4  — e2  u" )2 


. On  the  other  hand, 


r = , 

c«  d c^d 

the  triangle  spm  gives  sp  : pm  : : cos  l : sin  l ; that  is, 

ci*  c t r o \ • i a cos2  L 

— : -*/  ( a2  — x2 ) : : cos  l : sin  l : whence  x2= 

rf2  dw  v J ' dz—  eisin2 1. 

c/4  cosS  l 


And  from  a like  process  there  results,  tt2  = 


d2~e2sin2l 

Sub- 


92 


TRIGONOMETRICAL  SURVEYING. 


Substituting  in  the  equation  rot  = rji,  for  r,  and  r their 
values,  for  x 2 and  ua  their  values  just  found,  and  observing 
that  sin3  l -j-  cos2  l = 1,  and  sin2  J+cos2  / = 1,  we  shall  find 

OT  M 

•; J = •~3t 

(d2  — e2  sin2L)2  (d2  — e2  sin2  Z)T 

3.  , 2. 

or  m {d2  — e2  sin2  l)2  = m (d2  — e2  sin2  l)3  , 

or  wi3(d2  — e2  sin2  l)  = m®  (d2  — e2  sin2  l ). 

From  this  there  arises  e2  = d2  —c2  (by  hyp  ) = 

d3  (m3  — wfi  -d  . c2  d3 —c2 

i But,  __  t _ 

a . 2 . • d2  d2  ’ 

m3  sin2L— m3  sin2t 

and  consequently  the  reciprocal  of  this  fraction,  or 

2 2 _1  J_  X x . 

d2  M3sm2  l — ?»3sin2  l (M3sinL-f-  m3sin/)  . (M3sini.  — m3  sin  l) 


m3  cos il—  m 3 cos2  l (m3cos/-f- m3cos  l)  . (m3cos/— m3cos  l )■ 
Whence,  by  extracting  the  root,  there  results  finally 
X , X XX 

d , (M3sinL-f-m3sin  t)  . (M3sinL  — m3sin  l) 

c ^ x ’ I 7 I I ' 

(ra3COS?-j-M3COS  c)  . (m3cos/— m3cosl) 

This  expression,  which  is  simple  and  symmetrical,  has  been 
obtained  without  any  developement  into  series,  without  any 
omission  of  terms  on  the  supposition  that  they  are  indefinitely 
small,  or  any  possible  deviation  from  correctness,  except  what 
may  arise  frgm  the  want  of  coincidence  of  the  circle  of  cur- 
vature at  the  middle  points  of  the  arcs  measured,  with  the  arcs 
themselves  ; and  this  source  of  error  may  be  diminished  at 
pleasure,  by  diminishing  the  magnitude  of  the  arcs  measured 
though  it  must  be  acknowledged  that  such  a procedure  may 
give  rise  to  errors  in  the  practice,  which  may  more  than  coun- 
terbalance the  small  one  to  which  we  have  just  adverted. 

Cor.  Knowing  the  number  of  degrees,  or  the  parts  of  de- 
grees, in  the  measured  arcs  m,  to,  and  their  lengths,  which 
are  here  regarded  as  the  lengths  of  arcs  to  the  circles  which 
have  r,  r,  for  radii,  those  radii  evidently  become  known  in 
magnitude,.  At  the  same  time  there  are  given  the  algebraic 
values  of  r and  r ; thus,  taking  k for  example,  and  extermi- 

d5 

Dating  e2  andx2 , there  results  r= — — . There- 

c(d2 — (d2 — c-)  sin  x)^ 

fore,  by  putting  in  this  equation  the  known  ratio  of  d to  e, 
there  will  remain  only  one  unknown  quantity  d or  c,  which 
may  of  course  be  easily  determined  by  the  reduction  of  the 
last  equation  ; and  thus  all  the  dimensions  of  the  terrestrial, 
spheroid  will  become  known. 

General 


FIGURE  OF  THE  EARTH. 


93 


General  Scholium  and,  Remarks . 

1.  The  value — 1,  = — is  called  the  compression  of  the 

C C 0 

terrestrial  spheriod,  and  it  manifestly  becomes  known  when 

d 

the  ratio  - is  determined.  But  the  measurements  of  philoso- 

phers,  however  carefully  conducted,  furnish  resulting  com- 
pressions, in  which  the  discrepancies  are  much  greater  than 
might  be  wished.  General  Roy  has  recorded  several  of  these 
in  the  Phil  Trans,  vol.  77,  and  later  measurers  have  deduced 
others.  Thus,  the  degree  measured  at  the  equator  by 
Bouguer,  compared  with  that  of  France  measured  by  Me  ■ 

chain  apd  Delambre,  gives  for  the  compression  also 

d = 3271208  toises,  c = 3261443  toises,  d — c = 9765  toises. 
General  Roy’s  sixth  spheriod,  from  the  degrees  at  the  equa- 
tor and  in  latitude  45°,  gives  Mr.  Dal  by  makes  d — 

3489932  fathoms,  c = 3473656.  Col.  Mudge  d — 3491420, 
c = 3468007,  or  7935  and  7882  miles.  The  degree  mea- 
sured at  Quito,  compared  with  that  measured  in  Lapland  by 

Swanberg,  gives  compression  = — — — . Swanberg’s  observa- 

309*4  ^ 

tions,  compared  with  Bouguer’s  give  - — Swanberg’s 

compared  with  the  degree  of  Delambre  and  Mechain  — — . 

307'4 

Compared  with  Major  Lambton’s  degree  . A minimum 

of  errors  in  Lapland,  France,  and  Peru  gives Laplace, 

323*4 

from  the  lunar  motions,  finds  compression  =^.  From  the 
theory  of  gravity  as  applied  to  the  latest  observation  of  Burg, 
Maskelyne,  &c.  —q9,0~-  From  the  variation  of  the  pendulum 

in  different  latitudes  — *.  Dr.  Robinson,  assuming  the  va- 
335*78 

1 1 

riation  of  gravity  at  makes  the  compression  — 9*  Others 

give  results  varying  from  — — to  — : but  far  the  greater 
178-4  577 

number  of  observations  differ  but  little  from^,  which  the 

computation  from  the  phenomena  of  the  precession  of  the 
equinoxes  and  the  nutation  of  the  earth’s  axis,  gives  for  the 
maximum  limit  of  the  compression.  2.  From 

* This  number  3-3 .jj  does  not  result  from  the  variation  of  the  pen- 
dulum in  different  latitudes,  but  is  altogether  erroneous  in  conse- 
quence of  certain  numerical  mistakes  in  La  Place’s  calculations. 


94 


TRIGONOMETRICAL  SURVEYING. 


2.  From  the  various  results  of  careful  admeasurements  it 
happens,  as  Gen.  Roy  has  remarked,  “ that  philosophers  are 
not  yet  agreed  in  opinion  with  regard  to  tne  exact  hgure  of 
the  earth  ■ some  contending  that  it  has  no  regular  figure,  that 
is,  not  such  as  would  be  generated  by  the  revolution  of  a 
curve  around  its  axis.  Others  have  supposed  it  to  be  an 
ellipsoid  ; regular,  if  both  polar  sides  should  have  the  same 
degree  of  flatness  ; but  irregular  if  one  should  be  flatter  than 
the  other.  And  lastly,  some  suppose  it  to  be  a spheroid  dif- 
fering from  the  ellipsoid,  but  yet  such  as  would  be  formed  by 
the  revolution  of  a curve  around  its  axis.”  According  to  the 
theory  of  gravity  however,  the  earth  must  of  necessity  have 
its  axis  approaching  nearly  to  either  the  ratio  of  1 to  6b0  or 
303  to  304  : and  as  the  former  ratio  obviously  does  not  obtain, 
the  figure  of  the  earth  must  be  such  as  to  correspond  nearly 
with  the  latter  ratio. 

3.  Besides  the  method  above  described,  others  have  been 
proposed  for  determining  the  figure  of  the  earth  by  measure- 
ment. Thus  that  figure  might  be  ascertained  by  the  mea- 
surement of  a degree  in  two  parallels  of  latitude  ; but  not  so 
accurately  as  by  meridional  arcs,  1st.  Because,  when  the  dis- 
tance of  the  two  stations,  in  the  same  parallel  is  measured, 
the  celestial  arc  is  not  that  of  a parallel  circle,  but  is  nearly 
the  arc  of  a great  circle,  and  always  exceeds  the  arc  that  cor- 
responds truly  with  the  terrestrial  arc.  2dly,  The  interval 
of  the  meridian’s  passing  through  the  two  stations  must  be 
determined  by  a time-keeper,  a very  small  error  in  the  going 
of  which  will  produce  a very  considerable  error  in  the  com- 
putation. Other  methods  which  have  been  proposed,  are,  by 
comparing  a degree  of  the  meridian  in  any  latitude,  with  a 
degree  of  the  curve  perpendicular  to  the  meridian  in  the  same 
latitude  ; by  comparing  the  measures  of  degrees  of  the  curves 
perpendicular  to  the  meridian  in  different  latitudes  ; and  by- 
comparing  an  arc  of  a meridian  with  an  arc  of  the  parallel  of 
latitude  that  crosses  it.  The  theorems  connected  with  these 
and  some  other  methods  are  investigated  by  Professor  Play- 
fair in  the  Edinburgh  Transactions,  vol.  v,  to  which,  together 
with  the  books  mentioned  at  the  end  of  the  1st  section  of  this 
chapter,  the  reader  is  referred  for  much  useful  information 
on  this  highly  interesting  subject. 

Having  thus  solved  the  chief  problems  connected  with 
Trigonometrical  Surveying,  the  student  is  now  presented 
with  the  following  examples  by  way  of  exercise. 

Ex-  1.  The  angle  subtended  by  two  distant  objects  at  a 
third  object  is  66°  30'  39"  ; one  of  those  objects  appeared  un- 
der an  elevation  of  25'  47",  the  other  under  a depression  of  1 . 
Required  the  reduced  horizontal  angle.  Ans.  66°  30'  37  '. 

Ex.  2. 


FIGURE  OF  THE  EARTH. 


95 


Ex.  2.  Going  along  a straight  and  horizontal  road  which 
passed  by  a tower,  1 wished  to  find  its  height,  and  for  this 
purpose  measured  two  equal  distances  each  of  84  feet,  and  at 
the  extremities  of  those  distances  took  three  angles  of  eleva- 
tion of  the  top  of  the  tower,  viz.  36°  50',  21°  24’,  and  14°. 
What  is  the  height  of  the  tower  ? Ans.  53-96  feet. 

Ex.  3.  Investigate  General  Roy’s  rule  for  the  spherical  ex- 
cess, given  in  the  scholium  to  prob.  8. 

Ex.  4.  The  three  sides  of  a triangle  measured  on  the 
earth’s  surface  (and  reduced  to  the  level  of  the  sea)  are  17,  18, 
and  10  miles  : what  is  the  spherical  excess  ? 

Ex.  5.  The  base  and  perpendicular  of  another  triangle  are 
24  and  15  miles.  Required  the  spherical  excess. 

Ex.  6.  In  a triangle  two  sides  are  18  and  23  miles,  and 
they  include  an  angle  of  58°  24'  36".  What  is  the  spherical 
excess  ? 

Ex.  7.  The  length  of  a base  measured  at  an  elevation  of 
38  feet  above  the  level  of  the  sea  is  34286  feet : required  the 
length  when  reduced  to  that  level. 

Ex.  8.  Given  the  latitude  of  a place  48°  51'n,  the  sun’s 
declination  18°  30'n,  and  the  sun’s  altitude  at  10h  llm  26s  am, 
52°35'  ; to  find  the  angle  that  the  vertical  on  which  the  suu 
is,  makes  with  the  meridian. 

Ex.  9.  When  the  sun’s  longitude  is  29°  13'  43",  what  is 
his  right  ascension  ? The  obliquity  of  the  elliptic  being  23° 
27'  40". 

Ex.  10.  Required  the  longitude  of  the  sun,  when  his  right 
ascension  and  declination  are  32°  46  52"  i and  13°  13'  27".  n 
respectively.  See  the  theorems  in  the  scholium  to  prob.  12. 

Ex.  11.  The  right  ascension  of  the  star  a,  Ursas  majoris 
is  162°  50  34",  and  the  declination  62°  50'  n : what  are  the 
longitude  and  latitude  ? The  obliquity  of  the  ecliptic  being  as 
above. 

Ex.  12.  Given  the  measure  of  a degree  on  the  meridian  in 
n.  lat.  49°3',  60833  fathoms,  and  of  another  in  n.  lat.  12°32', 
60494  fathoms  : to  find  the  ratio  of  the  earth’s  axes. 

Ex.  13.  Demonstrate  that,  if  the  earth’s  figure  be  that  of 
an  oblate  spheroid,  a degree  of  the  earth’s  equator  is  the  first 
of  two  mean  proportionals  between  the  last  and  first  degrees 
of  latitude. 

Ex.  14.  Demonstrate  that  the  degrees  of  the  terrestrial 
meridian,  in  receding  from  the  equator  towards  the  poles,  are 

increased 


3d 


POLYGONOMETRY. 


increased  very  nearly  in  the  duplicate  ratio  of  the  sine  of  the 
latitude. 

jEx.  15.  If  p be  the  measure  of  a degree  of  a great  circle 
perpendicular  to  a meridian  at  a certain  point,  m that  of  the 
corresponding  degree  on  the  meridian  itself,  and  d the  length 
of  a degree  on  an  oblique  arc,  that  arc  making  an  angle  a 

with  the  meridian,  then  is  d ==  — — — --r  . ■ „ . Required  a 

Pt  (.m~pJ  s*n2  a ’ 

demonstration  of  this  theorem. 


PRINCIPLES  OF  POLYGONOMETRY. 

The  theorems  and  problems  in  Polygonometry  bear  an  in- 
timate connection  and  close  analogy  to  those  in  plane  trigo- 
nometry ; and  are  in  great  measure  deducible  from  the  same 
common  principles.  Each  comprises  three  general  cases. 

1.  A triangle  is  determined  by  means  of  two  sides  and  an 
angle  ; or,  which  amounts  to  the  same,  by  its  sides  except 
one,  and  its  angles  except  two.  In  like  manner,  any  rectili- 
near polygon  is  determinable  when  all  its  sides  except  one, 
and  all  its  angles  except  two,  are  known. 

2.  A triangle  is  determined  by  one  side  and  two  angles  ; 
that  is,  by  its  sides  except  two,  and  all  its  angles.  So  like- 
wise, any  rectilinear  figure  is  determinable  when  all  its  sides 
except  two,  and  all  its  angles,  are  known. 

3.  A triangle  is  determinable  by  its  three  sides  ; that  is 
when  all  its  sides  are  known  and  all  its  angles,  but  three.  In 
like  manner,  any  rectilinear  figure  is  determinable  by  means 
of  all  its  sides,  and  all  its  angles  except  three. 

In  each  of  these  cases,  the  three  unknown  quantities  may 
be  determined  by  means  of  three  independent  equations  ; the 
manner  of  deducing  which  may  be  easily  explained,  after  the 
following  theorems  are  duly  understood. 

THEOREM  I. 

In  Any  polygon,  any  One  Side  is  Equal  to  the  Sum  of  all 

The  Rectangles  of  Each  of  the  Other  Sides  drawn  into  the 

Cosine  of  the  Angle  made  by  that  Side  and  the  Proposed 

Side*. 


* This  theorem  and  the  following  one,  were  announced  by  Mr. 
Lexel  of  Petersburg,  in  Phil.  Trans,  vol.  65,  p.  282  : but  they  were 
first  demonstrated  by  Dr-  Hutton,  in  Phil.  Trans-  vol.  66,  pa.  600- 

Let 


POLYGONOMETRY.  97 

Let  abcdef  be  a polygon  : then 

will  AF  = AB  . COS  A + BC  . COS 
•gba  fa  4*  CD  ■ COS  cnA  AF  4-  DE  . cos 
DEA  AF  4-  EF  . COS  EFA  AF*. 

For,  drawing  lines  from  the  sever- 
al angles,  respectively  parallel  and 
perpendicular  to  af  ; it  will  be 

Ab  = AB  . COS  BAF, 
be  = B/J  = BC  . COS  CB/3  = BC  . COS  CBaAF, 

cd  = (J’d  = CD  . COS  CDiJ  = CD  . COS  CDA  AF, 

de  = £E  = DE  . COS  DEE  = DE  . COS  DEA  AF, 

eF  = ....  EF  . COS  EFe  = EF  . COS  EFA  AF.. 

But  af  — be  4"  cd  + de  + bf  — a6  ; and  a b,  as  expressed 
above,  is  in  effect  subtractive,  because  the  cosine  of  the  obtuse 
angle  baf  is  negative-  Consequently, 

AF  — AC  4”  cd-\-de  4"  eF  = ab  . cos  BAF  4"  BC  . cos  cba  af  4" 
&c.  as  in  the  proposition.  A like  demonstration  will  apply, 
mulatis  mutandis , to  any  other  polygon. 

Cor.  When  the  sides  of  the  polygon  are  reduced  to  three, 
this  theorem  becomes  the  same  as  the  fundamental  theorem 
in  chap,  ii,  from  which  the  whole  doctrine  of  Plane  Trigo- 
nometry is  made  to  flow. 

THEOREM  H. 

The  Perpendicular  let  fall  from  the  Highest  Point  or  Summit 
of  a Polygon,  upon  the  Opposite  Side  or  Base,  is  Equal  to 
the  Sum  of  the  Products  of  the  Sides  Comprised  between 
that  Summit  and  the  Base,  into  the  Sines  of  their  Respec- 
tive Inclinations  to  that  Base. 

Thus,  in  the  preceding  figure,  cc  ==  cb  . sin  ceafa4-b.a  . sin 
a i or  cc  = cd  . sin  cdaaf4*de  sin  deaaf-|“EF  . sin  f.  This 
is  evident  from  an  inspection  of  the  figure. 

Cor.  1.  In  like  manner  nd  — de  . sin  deaaf-{-ef  . sin  F, 
or  Dd  — cb  . sin  cbafa  4"  ba  sin  a — cd  . sin  cdaaf. 

Cor.  2.  Hence  the  sum  of  the  products  of  each  side,  into 
the  sine  of  the  sum  of  the  exterior  angles,  (or  into  the  sine  of 
the  sum  of  the  supplements  of  the  interior  angles),  comprised 
between  those  sides  and  a determinate  side,  is  = 4"  perp.  — 
perp.  or  = 0.  That  is  to  say,  in  the  preceding  figure, 
ab  . sin  a 4*  bc  . sin  (a  -f  b)  -j-  cd  . sin  (a4-b4-c)4"DE  . sin 
(a  4*  b 4"  c 4*  d)  4*  ef  . sin  (a  -(-  b 4"  c -f  d -f-  e)  = 0. 


* VVhen  a caret  is  put  between  iwb  letters  or  pairs  of  letters  deno- 
ting lines,  the  expression  altogether  denotes  the  angle  which  would  be 
made  by  those  two  lines  if  they  were  produced  till  they  met,  thus 
<E Baf  a denotes  the  inclination  of  the  line  cb  to  fa- 
Yob.  II.  14 


e F 


Here 


98 


POLYGONOMETRY. 


Here  it  is  to  be  observed,  that  the  sines  of  angles  greater  than 
180°  are  negative  (ch.  ii  equa.  vii). 

Cor,  3.  Hence  again,  by  putting  for  sin  (a+b),  sin  (a-}-b  + c), 
their  values  sin  a . cos  b -f  sin  b . cos  a,  sin  a . cos  (b  + c)  + 
sin  (b  -j-  c . cos  a,  &c.  (ch.  ii  equa.  v),  and  recollecting  that 

tang  = — (ch.  ii  p.  55),  we  shall  have, 

COS 

sin  A . (AB  -f-  BC  • COSB-f  CD  . cos(b-)-c)4-de  . cos('b  c + d)  + &c) 
+cosa.(bc  sinB-{-cD  . sin(B+c)+DE  . cos(B  + c-j-D)-{-&cj:=0  ; 
and  thence  finally,  tan  180°  — a,  or  tan  baf  = 

Be  . sinB+CD  • sin(B-f  c)  + de  . sin(B  + c4-D)  -i-  ef  . sin(B-f-  c + d + E) 
AB-t-BC-COSB-f  CD.C05(B-f-G)+DE.C',s(B  + C+D  J-f-EF.Cor- (B  + C-;  D + e) 

A similar  expression  will  manifestly  apply  to  any  polygon  ; 
and  when  the  number  of  sides  exceeds  four,  it  is  highly  useful 
in  practice. 

Cor . 4.  In  a triangle  abc,  where  the  sides  ab,  bc,  and  the 
angle  abc,  or  its  supplement  b,  are  known,  we  have 

, BC  . sin  b , AB  • sin  B 

tan  cab  = ; . . . . tan  bca  = : 

AB-J-BC  . COS  B BC  TAB  .COS  B 

in  both  which  expressions,  the  second  term  of  the  denomi- 
nator will  become  subtractive  whenever  the  angle  abc  is  acute, 
or  b obtuse. 

THEOREM  HI. 


The  Square  of  Any  Side  of  a Polygon,  is  Equal  to  the  Sum  of 
the  Squares  of  All  the  Other  Sides,  Minus  Twice  the  Sum 
of  the  Products  of  All  the  Other  Sides  Multiplied  two  and 
two,  and  by  the  Cosines  of  the  Angles  they  Include. 

For  the  sake  of  brevity,  let  the  sides 
be  represented  by  the  small  letters  which 
stand  against  them  in  the  annexed  figure  : \& 

then,  from  theor.  1,  we  shall  have  the  sub-  ~e\J \ 

joined  equations,  viz.  A.  & Ji 

a = b . cos  aAb  c . cos  aAc  + S . cos  aA  S, 

b = a . cos  aAb  -f-  c . cos  bsc  + S . cos  bAS, 

c — a . cos  cue.  -f-  b . cos  bAc  -f-  ^ • cos  cAS, 

S — a . cos  aAS  -f-  b . cos  b 'S  + c . cos  cA S. 

Multiplying  the  first  of  these  equations  by  a,  the  second  by  o 
the  third  by  c,  the  fourth  by  S ; subtracting  the  three  latter 
products  from  the  first,  and  transposing  b2 , c- , S2 , there  will 
result 

a2  = i2+c2+^2— 2(6c  . cos  bAc-\-bS • cos b'S -f-cJ1 . cos  cAS ). 
In  like  manner, 

c2  = a2+f>2-f  S2  — 2 (ab  . cos  aAJ-f  aS , cos  a-'S+bS . cos  b*S). 
&c.  &c. 


Or. 


POLYGONOMETRY. 


99 


Or,  since  bKc  = c , — c + d — 180°,  cAJ  = d,  we  have 

a2  — b2  + c2  4-  —2 (be  . cos  c — bS' . cos(c+i>)+c^.cos  d), 

c2  = a2  -j-  J2  _{_  ^3  — 2 (ab  . cos  b — b&  . cos(a4_b)+oJ' .cos  a). 
&c.  &c. 


The  same  method  applied  to  the  pentagon  abcde,  will  give 


a 2 =62 ^-c3  +J2  q-e 


2—2^' 


6c.cosc — bd  cos(c-f-n)  -J-6c.cos(c4-d+e) 

4 cd . cosd  — ce  . cos  (d4'e)  -f-  de  • cos  E 5 ’ 
And  a like  process  is  obviously  applicable  to  any  number  of 
sides  ; whence  the  truth  of  the  theorem  is  manifest. 

Cor.  The  property  of  a plane  triangle  expressed  in  equa.  i 
ch.  ii,  is  only  a particular  case  of  this  general  theorem. 


THEOREM  IV. 


Twice  the  Surface  of  Any  Polygon,  is  equal  to  the  sum  of 
the  Rectangles  of  its  Sides,  except  one,  taken  two  and  two, 
by  the  Sines  of  the  Sums  of  the  Exterior * Angles  Con- 
tained by  those  sides. 

1.  For  a trapezium,  or  polygon  of  four 
sides.  Let  two  of  the  sides  ab,  dc,  be 
produced  till  they  meet  at  p.  Then  the 
trapezium  abcd  is  manifestly  equal  to  the 
difference  between  the  triangles  pad  and 
pbc.  But  twice  the  surface  of  the  tri- 
angle pad  is  (Mens,  of  Planes  pr.  2 rule  2)  ap  . pd  . sin  p = 
(ab  4*  bp)  • (DC  cp)  . sin  p ; and  twice  the  surface  of  the 
triangle  pbc  is  = bp  . pc  . sin  p : therefore  their  difference, 
or  twice  the  area  of  the  trapezium  is  = (ab  . dc  + ab  . cp 
+ dc  . bp)  . sin  p.  Now,  in  A pbc, 


sin  p : sin  b : : bc  : pc,  whence  pc  = 
sin  p : sin  c : : bc  : pb,  whence  pb  = 


bc  • sin  b 
sin  p ’ 
BC  • sin  c 


Substituting  these  values  of  pb,  pc,  for  them  in  the  above 
equation,  and  observing  that  sin  p = sin  (pbc  + pcb)  ==  sin 
sum  of  exterior  angles  b and  c,  there  results  at  length, 


rrT  • c 1 ( AB  . BC  . Sill  B 

I wice  surface  / 1 

' ( -f-BC  . DC  . sin  c. 


of  trapezium. 


-j-AB  . dc  . sin  (b  c) 


Cor.  Since  ab  . bc  . sin  b = twice  triangle  abc,  it  follows 
that  twice  triangle  acd  is  equal  to  the  remaining  two  terms,  viz. 

twice  area  acd  = \ ab  . dc  . sin  (b  -f  c) 

( 4bc  . dc  . sin  c. 


* The  exterior  angles  here  meant,  are  those  formed  by  producing 
the  sides  in  the  same  manner  as  in  th.  20  Geometry,  and  in  cors.  1,  2, 
th*  2,  of  this  chap. 


2.  For 


106 


POLYGONOMETRY. 


2.  For  a pentagon,  as  abode.  Its  area 
is  obviously  equal  to  the  sum  of  the  areas 
of  the  trapezium  abcd,  and  of  the  tri- 
angle ade.  Let  the  sides  ab,  dc,  as  be- 
fore, meet  when  produced  at  p.  Then, 
from  the  above,  we  have 
Twice  area  of) 


AB 
+ AB 
+ BC 


the  trapezium  ) = 

ABCD  J 

And,  by  the  preceding  corollary, 
Twice  triangle  > _ ^ ap  . de 

DAE  $ 


BC 

DC 

DC 


Sin  E 

sin  (b  + c) 
sin  c. 


That  is  twice 
triangle  dae 

Now,  bp  = 


i- 


sin  (f+d)  or  sin  (b+c+ d} 
) + dp  . de  . sin  D. 

ab  . de  . sin  (b+c+d) 

+ DC  . DE  . sin  D 
+ BP  . DE  . sin  (b  + C + d) 

+ CP  . DE  . sin  D. 


sin  (B  +c)’ 
BC 


and  cp 


bc  . 


sin  (»  + c) 


s,n  % : therefore  the  last 


two  terms  become 


DE  . sin  c.sintB  -t-C+D)  BC.DE 
+ 


s:n  Cb  + c) 


BC.DE  , 


S)ii  (b  + c) 

sin  d +sin  c • sin  (b  -4-  c4-d}  , , . 

and  this  expression 


sin  (b  -f  c) 

by  means  of  the  formula  for  4 arcs  (art  30  ch.  iii,)  becomes 
bc  . de  . sin  (c+d).  Hence,  collecting  the  terms,  and  ar- 
ranging them  in  the  order  of  the  sides,  they  become 

ab  . bc  . sin  B 

+ AB  . DC 
I +AB  • de 
I + BC  . DC 
-f-BC  . DE 
+ DC  . DE 


Twice  the  area 
of  the  penta- 
gon ABCDE 


5- 


. sin  (b+c) 

. sin  (b+c+d) 
. sin  c 
. sin  (c+d) 

. sin  d. 


Cor.  Taking  away  from  this  expression,  the  1st,  2d,  and 
4th  terms,  which  together  make  double  the  trapezium  abcd 
there  will  remain 

Twice  area  of)  i ab.de.  sin  (b+c+d) 
the  triangle  > = < +bc  . de  . sin  (c+d) 

DAE  ) ( +DC  . DE  . Sin  D. 

3.  For  a hexagon,  as  abcdef.  The 
double  area  will  be  found,  by  supposing 
it  divided  into  the  pentagon  abcde,  and 
the  triangle  aef.  For,  by  the  last  rule, 
and  its  corollary,  we  have, 


P C 


Twice 


POLYGONOMETRY. 


301 


Twice  area  of  1 
the  pentagon  \ 
ABODE  5 


Twice  area  of} 
the  triangle  > 


r ab 

| +AB 

J +AB 

1 +BC 
| +BC 

L+cr> 

AF 
+ DP 
+ DE 


BC 

CD 

DE 

CD 

DE 

DE 

EF 

EF 

EF 


Or,  twice  area  of 
the  trian, 

AEF 


L Ofi 

gle\ 


EF 

EF 


f AB 

I -f-DC 

= +OE  . EF 
| +BP  . EF 
1.+CP  . EF 


sin  B 

sin  (s-j-c) 
sin  (b-{-c+d) 
sin  c 

sin  (c+d) 
sin  d. 

sin  (b+c  + d-J-e) 
sin  (d+e) 

, sin  e. 

sin  (b+c+d+e) 
sin  (d+e) 
sin  e 

, sin  (b+c+d+e) 
, sin'(D-j-E). 


Now,  writing  for  bp,  cp,  their  respective  values, 

BC  sin  c , BC  , sin  b „ 

- — t — : — -and  — the  sum  of  the  last  two  expressions 

sin  (b+c)  sm(B-J-c) 

jn  the  double  areas  of  aef,  will  become 

sin  c sin  (B+c  + D+E)+sin  B • s'n  (D-f-E'i 


BC  • EF 


Twice  the  area 
of  the  hexa 

son  ABCDEF 


■h 


sin  (b  + c) 

and  this,  by  means  of  the  formula  for  5 arcs  (art.  30  ch.  iii) 
becomes  bc.  ef  sin  (c-f-D+E).  Hence,  collecting  and  pro- 
perly arranging  the  several  terms  as  before,  we  shall  obtain 

ab  . bc  . sin  B 

+ AB  . CD 
-f-AB  . DE 
+ AB  . EF 
-{-BC  . CD 
+ BC  . DE 
+ BC  . EF 
-(-CD  . DE 
-{-CD  . EF 
t + DE  . EF 

4.  In  a similar  manner  may  the  area  of  a heptagon  be  de 
termined,  by  finding  the  sum  of  the  areas  of  the  hexagon  and 
the  adjacent  triangle  : and  thence  the  area  of  the  octagon, 
nonagon,  or  of  any  other  polygon,  may  be  inferred  ; the  lawT 
of  continuation  being  sufficiently  obvious  from  what  is  done 

above,  and  the  number  of  terms  = - — - . - — , when  the  num- 

12 

ber  of  sides  of  the  polygon  is  n : for  the  number  of  terms  is 
evidently  the  same  as  the  number  of  ways  in  which  n — 1 quan- 
tities can  be  taken,  two  and  two  ; that  is,  (by  the  nature  of 

Permutations)  = ?^ 1 . 2. 


sin  (b+c) 
sin  (b+c+d) 

$iu  (B-j-C-f-D-j-E) 
sin  c 

sin  (c+d) 
sin  (c+d+e) 
sin  d 

sin  (d-(-e) 

, sin  e 


Scholium. 


102 


POLYGONOMETRY. 


Scholium. 

This  curious  theorem  was  first  investigated  by  Simon  Lhuil • 
Her,  and  published  in  1789.  Its  principal  advantage  over  the 

common  method  for  finding  the  areas  of  irregular  polygons 

is,  that  in  this  method  there  is  no  occasion  to  construct  the 
figures,  and  of  course  the  errors  that  may  arise  from  such 
constructions  are  avoided. 

In  the  application  of  the  theorem  to  practical  purposes,  the 
expressions  above  become  simplified  by  dividing  any  proposed 
polygon  into  two  parts  by  a diagonal,  and  computing  the  sur- 
face of  each  part  separately. 

Thus,  by  dividing  the  trapezium  abcd  into  two  triangles, 
by  the  diagonal  ac,  we  shall  have- 

Twice  area  ^ __y  C ab  . bc  . sin  b 

trapezium  } \ +cd  . ad  . sin  d. 

The  pentagon  abcde  may  be  divided  into  the  trapezium 
abcd,  and  the  triangle  ade,  whence 

P ab  . bc  . sin  B 

Twice  area  of  ? __  ) -f  ab  . dc 

pentagon  3 \ +bc  . do 

( -f-r>E  . ae 

Thus  again,  the  hexagon  may  be  divided  into  two  trape- 
ziums, by  a diagonal  drawn  from  a to  d,  which  is  to  be  the 
line  excepted  in  the  theorem  ; then  will 
ab  . bc  . sin  B 


sin  (b  + c) 
sin  c 
sin  e. 

be  divided  into  two 


Twice  area  of  1 
hexagon  { 


J 


+ AB 
__  J +BC 
+ DE 

+de 

+ EF 


DC 

DC 

EF 

AF 

AF 


sin  (b+c) 
sin  c 
sin  e 

sin  (e+f) 
sin  f. 


And  lastly,  the  heptagon  may  be  divid- 
ed into  a pentagon  and  a trapezium,  the 
diagonal,  as  before,  being  the  excepted 
line  : so  will  the  double  area  be  expressed 
by  9 instead  of  15  products,  thus  : 


Twice  area 
heptagon 


of 


H 


AB 
+ AB 
+ AB 
+ EC 
+ BC 
+ CD 
+ EF 
+ EF 
L+FG 


BC 

CD 

DE 

CD 

DE 

DE 

FG 

GA 

GA 


Sill  B 

sin  (b+c) 
sin  (b+c+d) 
sin  c 

sin  (c+d) 
sin  d 
sin  f 

sin  (f+g) 
sin  g. 


The  same  method  may  obviously  be  extended  to  other  poly- 
gons, with  great  ease  and  simplicity. 

It 


POLYGONOMETRY. 


J03 


It  often  happens,  however,  that  only  one  side  of  a polygon 
can  be  measured,  and  the  distant  angles  be  determined  by  in- 
tersection ; in  this  case  the  area  may  be  found,  independent 
of  construction,  by  the  following  problem. 

PROBLEM  I. 

Given  the  Length  of  One  of  the  Sides  of  a Polygon,  and  the 
Angles  made  at  its  two  extremities  by  that  Side  and  Lines 
drawn  to  all  the  Other  Angles  of  the  Polygon  : to  find  an 
Expression  for  the  Surface  of  that  Polygon. 

Here  we  suppose  known  pr  ; also 

APR  = a',  BPR  =a=  b' , CPR  = C , DPR  = d' ; 

ARP  = a",  BRP  = b",  CRP  ==  c',  DRP  = d" . 

Now,  sin  par  ==  sin  (a'+a")  i sin  pbr  = 
sin  (6'-f-Y'). 


sin  a 


PC. 


Therefore,  sin  (a'  + «")  : PR  : : sin  a"  : pa  = 

sin  (a  -}-  a") 
sin  b'' 

And,  ...  sin  (6'  + 6")  : pr  : : sin  b''  : pb  pr. 

sin(6'+6") 

ap  . pb  . i sin  apb  = a ap  . pb  . sin  (a'  — b ) . 

sin  a"  . sin  b"  . sin  ( a'  — b's) 

a fr2— : — . 

sin  (a' + a")  . sin  (4'  4~  b ') 
sin  b"  . sin  c" . sin  ( b'  — c) 


But,  triangle  apb 
Hence,  surface  A apb 


In  like  manner,  A bpc  = 4 pR2 


A cpd  = 4 PR2 


sin  (6’4- b")  . sin  (c  4~  c") 
sin  c"  . sind''.  sin  (c'  — d! ) 


AdPr 


sin(c'  4-  c'1)  . sin  (cT+  d") 
&c.  &c.  &c. 

sin  d" 

rp  . pd  . |sin  dpr  = pr. .ApQ  . gin  d"  = 


Aprs  . 


sin  d!  . sin  d" 


sin  (d'  -f-  d') 


sin  (cZ'-fd") 
Consequently, 


sin  a"  . sin  b"  . sin  (a'  — b' ) 


sin  (a  4-  a")  . sin  (6'  -f  b") 
sin  b‘‘  . sin  c"  . sin  ( b'  — c) 

Surface  pabcdr  = 4pr2.  J s!n  ^ ) ' s*n  (c  c') 

Sin  c . sin  d''  . sin  ( r'  


+ 


4- 


sin  c"  . sin  d''  . sin  (c'  — d‘ ) 

sin  (c  4"  c")  . sin  ( d ' 4-  d") 
sin  d'  . sin  d' 


sin  (d'  d") 


The 


104 


POLYGONOMETRY. 


The  same  method  manifestly  applies  to  polygons  of  aur 
number  of  sides  : and  all  the  terms  except  the  last  are  so  per- 
fectly symmetrical,  while  that  last  term  is  of  so  obvious  a form, 
that  there  cannot  be  the  least  difficulty  in  extending  the  formula 
to  any  polygon  whatever. 

PROBLEM  II. 


Given,  in  a Polygon,  All  the  Sides  and  Angles,  except  three  ; 
to  find  the  unknown  Parts. 


This  problem  may  be  divided  into  three  general  cases,  as 
shown  at  the  beginning  of  this  chapter  : but  the  analytical 
solution  of  all  of  them  depends  on  the  same  principles  ; and 
these  are  analogous  to  those  pursued  in  the  analytical  investi- 
gations of  plane  trigonometry  In  polygonometry,  as  well  as 
trigonometry,  when  three  unknown  quantities  are  to  be  found, 
it  must  be  by  means  of  three  independent  equations,  involving 
the  known  and  unknown  parts.  These  equations,  may  be  de- 
duced from  either  theorem  1,  or  3,  as  may  be  most  suited  to 
the  case  in  hand  ; and  then  the  unknown  parts  may  each  be 
found  by  the  usual  rules  of  e> termination. 

For  an  example,  let  it  be  supposed  that  p 

in  an  irregular  hexagon  abcdef,  there  are  E, .. 

given  all  the  sides  except  ab,  bc,  and  all  / 

the  angles  except  b ; to  determine  those  / 

three  quantities.  L \ / 

-A.  B 


The  angle  b is  evidently  equal  to  (2 n — 4)  right  angles  — 
(a  + c -j-  d -j-  e -f-  f)  ; n being  the  number  of  sides,  and  the 
angles  being  here  supposed  the  interior  ones. 

Let  ab  = x,  bc  = y : then  by  th.  1, 
x — y . cos  b -f-  DC  . cos  abacd  -j-  de  . cos  abaed 

+ EF  . COS  ABAEF  -f-  AF  . COS  ABaAF  ; 

1J  = a-  . COS  B + AF  . COS  BCaAF  + FE  . COS  BCA  FE. 

+ DE  . COS  BCADE  + DC  . COS  BCACD. 

In  the  first  of  the  above  equations,  let  the  sum  of  all  the 
terms  after  y . cos  b,  be  denoted  by  c ; and  in  the  second  the 
sum  of  all  those  which  fall  after  x . cos  b,  by  d ; both  sums 
being  manifestly  constituted  of  known  terms  : and  let  the 
known  coefficients  of  x and  y be  in  and  n respectively.  Then 
will  the  preceding  equations  become 

x ~ ny  -f-  c . . . . y = mx  -j-  d. 

Substituting  for  y,  in  the  first  of  the  two  latter  equations,  its 
value  in  the  second,  we  obtain  x — mnx  + nd  -j-  c.  Whence 
there  will  readily  be  found 


POLYGONOMETRY. 


105 


Thus  ab  and  bc  are  determined.  Like  expressions  will  serve 
for  the  determination  of  any  other  two  sides,  whether  conti- 
guous or  not  : the  coefficients  of  x and  y being  designated  by 
different  letters  for  that  express  purpose  ; which  would  have 
been  otherwise  unnecessary  in  the  solution  of  the  individual 
case  proposed. 

Remark.  Though  the  algebraic  investigations  commonly 
lead  to  results  which  are  apparently  simple,  yet  they  are  often, 
especially  in  polygons  of  many  sides,  inferior  in  practice  to 
the  methods  suggested  by  subdividing  the  figures.  The  fol- 
lowing examples  are  added  for  the  purpose  of  explaining  those 
methods  : the  operations  however  are  merely  indicated  ; the 
detail  being  omitted  to  save  room. 


EXAMPLES. 


Ex.  1.  In  a hexagon  abcdef,  all  the  sides  except  af,  and 
all  the  angles  except  a and  f,  are  known.  Required  the  un- 
known parts.  Suppose  we  have 


ab  = 1284  Ext.  ang.  Whence 

bc  = 1782  b = 32°  b +c  = 80° 

cd  = 2400  c «=  48°  b 4-  c -+  d = 132° 

DE  = 2700  d = 52°  B + C 4- b -f- e = 198° 

EF  = 2860  E = 66°  A 4-  f = 162°. 

Then,  by  cor.  3 th.  2,  tan  baf  = 

BC  . sin  B 4 CD  . si  l(B  t-c  ;4-De  . sin(B  + C+D)  + EF.sill(B  + C+D+E) 
ab+bc  cu^b  + cd.cos(b4-c)-1-de.cos(.  b4c+d)+ef.cos(b+ct  d + e) 

Bc , ?in  32°  -f-  CD  • sin  80°  + de  ■ sin  132°  + Bf  • sin  198° 

ab+bc  . c js32°+cd  • cos  80^+0 E . cos  132°+ef  . cos  198" 
BC  • sin  32°  + CD  . sin  80e  + de.  sin  48°  — Ef  . sin  18° 

~ ab  v bc  . cos  32°4-cd  . co^  80°  — de  . cos  48°—  ef  . cos  18° 


Whence  baf  is  found  106°  31'  38"  ; and  the  other  angle  afe  = 
91°28'22".  So  that  the  exterior  angles  a and  f are  73°28'22'', 
and  88°31'38'’  respectively  : all  the  exterior  angles  making  4 
right  angles,  as  they  ought  to  do  Then,  all  the  angles  being 
known,  the  side  af  is  found  by  th.  1 = 4621  5. 

If  one  of  the  angles  had  been  a re-entering  one,  it  would 
have  made  no  other  difference  in  the  computation  than  what 
would  arise  from  its  being  considered  as  subtractive. 


Ex.  2.  In  a hexagon  abcdef,  all  the  sides  except  af,  and 
all  the  angles  except  c and  b,  are  known  : viz. 

Vox . II.  15  jlS 


106 


POLYGONOMETRY. 


ab=2400  Ex.  Ang. 
bg=2700  a=54° 

cd=3200  b=62° 

de=3500  e=64° 

ef=3760  f=72° 


We  shall  have,  by  th.  2 cor  1, 


ab  . sin  a 
-{-bc  . sin  (a+b) 
+cd  . sin  (a-}-b+c) 


!»  be. sin  ( 

t+,F- 


(e+f) 

sin  f. 


Therefore,  cd  . sin  (116°  -f-  c)  = 


— ab  . sin  54° 
— bc  . sin  116° 
+de  . sin  136° 
+ef  . sin  72° . 


Or  1 16°  4-  c = $ 149°23'26", 

’ + C J + 33°3H'34" 

The  second  of  these  will  give  for  c,  a re-entering  angle 
the  first  will  give  exterior  angle  c = 33c  23'  26'',  and  then 
will  d = 14°  36'34''.  Lastly, 
cos  54° 
cos  64° 

cos  30°36'34"  > = 3885-905. 
cos  44° 
cos  72° 

. Ex.  3.  In  a hexagon  abcdef,  are  known,  all  the  sides  ex- 
cept af,  and  all  the  angles  except  b and  e ; to  find  the  rest. 

Given  ab  = 1200  Exterior  angles  a = 64° 
bc  = 1500 

cd  = 1600  c = 72^ 

de  = 1800  d = 75° 

ef  — 2000 


f = 84°. 

Suppose  the  diagonal  be  drawn,  dividing  the  figure  into  two 
trapeziums.  Then,  in  the  trapezium  bcde  the  sides,  except 
be,  and  the  angles  except  b and  e,  will  be  known  ; and  these 
may  be  determined  as  in  exam.  1.  Again,  in  a trapezium 
abf.f,  there  will  be  known  the  sides  except  af,  and  the 
angles  except  the  adjacent  ones  b and  e.  Hence,  first  for 
bcde  : (cor.  3 th.  2), 

Cd  . sin  C + de  . sin  (c+d) 

tan  cbe  = — , a- — — -= 

BC  + CD  • COS  C-f-DE  . C0s(c  •+■  D) 

CD  . sin  72°  4-  de  . sin  147°  CD  . sin  72°-f-DE  . sin  S3° 

BC-bcD  . cos72°4-de  . cos  147°  bc  -f-  CD  . cos  72s  — de  . cos  35* 
Whence  cbe  = 79°  2'  1"  ; and  therefore  deb  = 67°57'  59'. 

!bc  . cos  79°  2 ' 1"  i 
+ cd  . cos  7°  2'  1'' > = 2548-581. 

4-de  . cos67°57'59  j 
Secondly,  in  the  trapezium  abef, 
ab  . sin  a + be  . sin  (a  + b)  = ef  . sin  f : whence 

. . , . ef  • sin  f — ab  . sin  b . i 20°55'54  , 
S!n(A  + B)=: =«in  j169,  4,  g. 

Taking 


POLYGONOMETRY. 


107 


Taking  the  lower  of  these  to  avoid  re-entering  angles,  we 
have  b ("exterior  ang.)  = 95°  4 ' 6"  ; abe  = 84°  55'  54  ' ;feb  =j 
63®  4'6''  : therefore  abc  = 163®  57'  55' ; and  fed  = 131°2’5" : 
aDd  consequently  the  exterior  angles  at  b and  e are  16°  2'  b' 
and  48®  57'  55'’  respectively. 

Lastly,  af  = — ab  . cos  a — be  . cos  (a  + b)  — ef  cos  f = — 
ab  . cos  64®  + be  . cos  20°  55'  54"  — ef  . cos  84®  = 1645-292. 

Nett.  The  preceding  three  examples  comprehend  all  the 
varieties  which  can  occur  in  Polygonometry,  when  all  the  sides 
except  one,  and  all  the  angles  but  two,  are  known.  The  un- 
known angles  may  be  about  the  unknown  side  ; or  thej'  may 
be  adjacent  to  each  other,  though  distant  from  the  unknown 
side  ; and  they  may  be  remote  from  each  other,  as  well  as 
from  the  unknown  side. 

Ex.  4.  In  a hexagon  abcdef,  are  known  all  the  angles,  and 
. all  the  sides  except  af  and  cd  : to  find  those  sides. 

Given  ab  = 2200  Ext.  Ang.  a = 96° 
bc  = 2400  b = 54® 

c = 20° 

de  = 4800  d = 24° 

ef  = 5200  e = 18° 

f = 148®. 


Here,  reasoning  from  the  principle  of  cor.  th.  2,  we  have/1 
AB  sin  96®")  C . , ,-,0o’  ab  sin  84®")  r . . .- 

+ BC  . sin  150® i — < s IL+  bc  sin  3b°i  = \ de.sui14° 

-f  cd  . sin  170®5  C +E  ■ Sm  148  + cn.sinl0°5  (+EF-Sinj2 

Whence  C DE.sinl4®  coseclO®  — A b. sin  84°.  cosec  10°  } 

Cd=  £ 4-EF.sm32°-coseclO® — bc  sin  30®  cosec  10®  c=°045-58. 

A 1 1 AsO  1 AO  ■ . onn  « 


And 
AF 


l 


DE  sin24°.cosecl0® — CB  sin  20®  ' 


3®  7 

+EF  sin42°.cosecl0° — BA.s'n74°y  ~ 148r4-98- 
Ex.  5.  In  the  nonagon  abcdefchi,  all  the  sides  are  known 
and  all  the  angles  except  a,  d,  g : it  is  required  to  find  those 
angles. 

Given  ab  = 2400  fg  = 3800  Ext.’  ang.  b = 40® 


bc  = 2700  gh  — 4000 
cd  = 2800  hi  ==  4200 
de  = 3200  IA  = 4500 
ef  = 3500 

Suppose  diagonals  drawn  to  join  the 
Unknown  angles,  and  dividing  the  po- 
lygon into  three  trapeziums  and  a tri- 
angle ; as  in  the  marginal  figure.  Then, 
1st.  In  the  trapezium  abcd,  where 
ad,  and  the  angles  about  it  are  unknown 
we  have  (cor.  3.  th.  2) 


c = 32° 
e = 36° 
f = 45° 
h = 48° 
i = 50®. 


tan 


108 


POLYGONOMETRY. 


. BC.s>nB-f-CD.sinfB+c1  BC.sin40f>+ CDsir>70* 

tan  bad— v-  --  = 

AB  f BC  CC-E4-CD  COs(B-f  c)  AB  + BC.COS40°+  CD.COS72® 

Whence  bad  = 39°  30‘  42",  cda  = 32°  29  18". 

!ab  . cos  39°  30  42"  i 
+bc  . cos  0 29  18  \ = 6913-292. 

4-cd  . cos  32  29  18  ) 

2dly.  In  the  quadrilateral  defg,  where  dg  and  the  angles 
about  it  are  unknown  ; we  have 

. EF.sinE-t*  f G.sinf e*4- f)  EF.sin36°+FG  S'h81° 

tan  edg  = - — - — = — n — o 

DE  EF.COSE  -J-  FG.COS(E+F)  DF.+  EF.C0S36o-f  FG-COSOl 

Whence  edg  = 41°  14'  53",  fgd  = 39°  45  7". 
de  . cos  41°  14' 


And  dg  — 


+ EF 
+ FG 


cos  5°  14 
cos  39®  45 


4'  53"  ) 

4'  53''  } = 
5'  7"  ) 


8812-803. 


3dly.  In  the  trapezium  ghia,  an  exactly  similar  process  gives 
HGA  = 50°  46'  53",  iag  •=  47°  13'  7",  and  ag  = 9780-591. 

4thly.  In  the  triangle  adg,  the  three  sides  are  now  known, 
to  find  the  angles  : viz  dag  = 60°  53'  26",  agd  = 43°  15'  54", 
adg  = 75°  50  40".  Hence  there  results,  lastly, 

IAB  =47°  13>  7''4-60°53'26''+S9°  30'42"  = 147°  37'  15", 
cde  =32°  29'18"+70°50'40''+41°  14  53''=  149°  34'  51'  , 
fgh=39°  45'  7''+43o15'54"+50°  46  53"  = 133°  47'  54". 
Consequently,  the  required  exterior  angles  are  a=32°22'45'  , 
j>  = 30°  25"  9",  g = 46°  12'  6". 

Ex.  6.  Required  the  area  of  the  hexagon  in  ex.  1. 

Ans.  16530191. 

Ex.  7.  In  a quadrilateral  abcd,  are  given  ab=24,  bc  = 30, 
cd  = 34  : angle  aec  = 92°  18',  bcd  = Q 7°  23'.  Required  the 
side  ad,  and  the  area. 


Ex.  8.  In  prob.  1,  suppose  pq  = 2538  links,  and  the  angles 
as  below  ; what  is  the  area  of  the  field  abcd^p  ? 

apq=89°  14',  bfq=68°  11',  cpq=36°  24',dpq=  19°  57  ; 
aqp=25°  18',  bqp=69°  24',  c<*p=94q  6',d$p=121°  18'. 


OF 


[ 109  3 


OF  MOTION,  FORCES,  &c. 

DEFINITIONS. 

Art.  1.  BODY  is  the  mass,  or  quantity  of  matter,  in  any 
material  substance  : and  it  is  always  proportional  to  its  weight 
or  gravity,  whatever  its  figure  may  be. 

2.  Body  is  either  Hard,  Soft,  or  Elastic.  A Hard  Body 
is  that  whose  parts  do  not  yield  to  any  stroke  or  percussion, 
but  retains  its  figure  unaltered.  A Soft  Body  is  that  whose 
parts  yield  to  any  stroke  or  impression,  without  restoring 
themselves  again  ; the  figure  of  the  body  remaining  altered. 
And  an  Elastic  Body  is  that  whose  parts  yield  to  any  stroke, 
but  which  presently  restore  themselves  again,  and  the  body 
regains  the  same  figure  as  before  the  stroke. 

We  know  of  no  bodies  that  are  absolutely,  or  perfectly, 
either  hard,  soft,  or  elastic  ; but  all  partaking  these  proper- 
ties, more  or  less,  in  some  intermediate  degree. 

3.  Bodies  are  also  either  Solid  or  Fluid.  A Solid  Body, 
is  that  whose  parts  are  not  easily  moved  among  one  another, 
and  which  retains  any  figure  given  to  it.  But  a Fluid  Body' 
is  that  whose  parts  yield  to  the  slightest  impression,  being 
easily  moved  among  one  another  ; and  its  surface,  when  left 
to  itself,  is  always  observed  to  settle  in  a smooth  plane  at 
the  top. 

4.  Density  is  the  proportional  weight  or  quantity  of 
matter  in  any  body'.  So,  in  two  spheres,  or  cubes,  &c.  of 
equal  size  or  magnitude  ; if  the  one  weigh  only  one  pound, 
but  the  other  two  pounds  ; then  the  density  of  the  latter  is 
double  the  density  of  the  former  ; if  it  weigh  3 pounds,  its 
density  is  triple  ; and  so  on. 

5.  Motion  is  a continual  and  successive  change  of  place.— 
If  the  body  move  equally,  or  pass  over  equal  spaces  in  equal 
times,  it  is  called  Equable  or  Uniform  Motion.  But  if  it 
increase  or  decrease,  it  is  Variable  Motion  ; and  it  is  called 
Accelerated  Motion  in  the  former  case,  and  Retarded  Motion 
in  the  latter. — Also,  when  the  moving  body  is  considered 

with 


110 


OF  MOTION,  FORCES,  &c. 


with  respect  to  some  other  body  at  rest,  it  is  said  to  be  Ab- 
solute Motion.  But  when  compared  with  others  in  motion, 
it  is  called  Relative  Motion. 

6.  Velocity,  or  Celerity,  is  an  affection  of  motion,  by 
which  a body  passes  over  a certain  space  in  a certain  time. 
Thus,  if  a body  in  motion  pass  uniformly  over  40  feet  in 
4 seconds  of  time,  it  is  said  to  move  with  the  velocity  of  10 
feet  per  second  ; and  so  on. 

7.  Momentum,  or  Quantity  of  Motion,  is  the  power  or 
force  in  moving  bodies,  by  which  they  continually  tend  from 
their  present  places,  or  with  which  they  strike  any  obstacle 
that  opposes  their  motion. 

8.  Force  is  a power  exerted  on  a body  to  move  it,  or  to 
stop  it.  If  the  force  act  constantly,  or  incessantly,  it  is  a 
Permanent  Force  : like  pressure  or  the  force  of  gravity. 
But  if  it  act  instantaneously,  or  but  for  an  imperceptibly 
small  time,  it  is  called  Impulse,  or  Percussion  : like  the  smart 
blow  of  a hammer. 

9.  Forces  are  also  distinguished  into  Motive,  and  Accele- 
rative or  Retarding.  A Motive  or  Moving  Force,  is  the 
power  of  an  agent  to  produce  motion  ; and  it  is  equal  or 
proportional  to  the  momentum  it  will  generate  in  any  body, 
when  acting,  either  by  percussion,  or  for  a certain  time  as  a 
permanent  force. 

10.  Accelerative,  or  Retardive  Force,  is  commonly  un- 
derstood to  be  that  which  affects  the  velocity  only  ; or  it  is 
that  by  which  the  velocity  is  accelerated  or  retarded  ; and  it 
is  equal  or  proportional  to  the  motive  force  directly,  and  to 
the  mass  or  body  moved  inversely. — So,  if  a body  of  2 pounds 
weight,  be  acted  on  by  a motive  force  of  40  ; then  the 
accelerating  force  is  20.  But  if  the  same  force  of  40  act  on 
another  body  of  4 pounds  weight  ; then  the  accelerating 
force  in  this  latter  case  is  only  10  ; and  so  is  but  half  the 
former,  and  will  produce  only  half  the  velocity. 

11.  Gravity,  or  Weight,  is  that  force  by  which  a body 
endeavours  to  fall  downwards.  It  is  called  Absolute  Gravity.  _ 
when  the  body  is  in  empty  space  ; and  Relative  Gravity, 
when  emersed  in  a fluid. 

12.  Specific  Gravity  is  the  proportion  of  the  weights  ot 
different  bodies  of  equal  magnitude  ; and  so  is  proportional 
to  the  density  of  the  body. 


AXrOMS 


GENERAL  LAWS  OF  MOTION. 


Ill 


AXIOMS. 

13.  Every  body  naturally  endeavours  to  continue  in  its 
present  state,  whether  it  be  at  rest,  or  moving  uniformly  in 
a right  line. 

14.  The  Change  or  Alteration  of  Motion,  by  any  external 
force,  is  always  proportional  to  that  force,  and  in  the  direction 
of  the  right  line  in  which  it  acts. 

15.  Action  and  Re-action,  between  any  two  bodies,  are 
equal  and  contrary.  That  is,  by  Action  and  Re-action,  equal 
changes  of  motion  are  produced  in  bodies  acting  on  each  oth- 
er ; and  these  changes  are  directed  towards  opposite  or  con- 
trary parts. 


>♦< 


GENERAL  LAWS  OF  MOTION,  &c. 

PROPOSITION  L 

16.  The  Quantity  of  Matter,  in  all  bodies,  is  in  the  Compound 
Ratio  of  their  Magnitudes  and  Densities. 

That  is,  b is  as  md  ; where  b denotes  the  body  or  quantity 
of  matter,  m its  magnitude,  and  d its  density. 

For,  by  art.  4,  in  bodies  of  equal  magnitude,  the  mass  or 
quantity  of  matter  is  as  the  density.  But,  the  densities  re- 
maining, the  mass  is  as  the  magnitude  : that  is,  a double  mag- 
nitude contains  a double  quantity  of  matter,  a triple  magnitude 
a triple  quantity,  and  so  on.  Therefore  the  mass  is  in  the 
compound  ratio  of  the  magnitude  and  density. 

17.  Corol.  1.  In  similar  bodies,  the  masses  are  as  the  den- 
sities and  cubes  of  the  diameters,  or  of  any  like  linear  dimen- 
sions.— For  the  magnitudes  of  bodies  are  as  the  cubes  of  the 
diameters,  &c. 

18.  Corol.  2.  The  masses  are  as  the  magnitudes  and  specific 
gravities. — For,  by  art.  4 and  12,  the  densities  of  bodies  are 
as  the  specific  gravities. 

19.  Scholium.  Hence,  if  b denote  any  body,  or  the  quan- 
tity of  matter  in  it,  m its  magnitude,  d.  its  density,  g its 

• ' ■ specific 


112 


OF  MOTION,  FORCES,  &c. 


specific  gravity,  and  a its  diameter  or  other  dimension  ; then, 
oc  (pronounced  or  named  as)  being  the  mark  for  general 
proportion,  from  this  proposition  and  its  corollaries  we  have 
these  general  proportions  : 


b oc 

md 

oc 

mg 

oc 

a 3d, 

b 

b 

m oc 

- 

oc 

- 

a 

a3, 

d 

g 

b 

mg 

d oc 

- 

a 

g 

oc 

m 

a3 

b 

mg 

a3  oc 

— 

a 

m 

oc 

— . 

d 

d 

PROPOSITION 

20.  The  Momentum , or  Quantity  of  Motiony  generated  by  a 
Single  Impulse,  or  any  Momentary  Force,  is  as  the  Generating 
Force. 

That  is,  tn  is  as  f ; where  m denotes  the  momentum,  and 
f the  force. 

For  every  effect  is  proportional  to  its  adequate  cause.  So 
that  a double  force  will  impress  a double  quantity  of  mo- 
tion ; a triple  force,  a triple  motion  ; and  so  on.  That  is. 
the  motion  impressed,  is  as  the  motive  force  which  pro- 
duces it. 

PROPOSITION  ni. 

21.  The  Momenta,  or  Quantities  of  Motion,  in  moving  Bodies, 
are  in  the  Compound  Ratio  of  the  Alasses  and  Velocities. 

That  is,  m is  ae  bv. 

For,  the  motion  of  any  body  being  made  up  of  the  mo- 
tions of  all  its  parts,  if  the  velocities  be  equal,  the  momenta 
will  be  as  the  masses  ; for  a double  mass  will  strike  with  a 
doubleforce;  a triple  mass,  with  a triple  force,  and  soon. 
Again,  when  the  mass  is  the  same,  it  will  require  a double 
force  to  move  it  with  a double  velocity,  a triple  force  with  a 
triple  velocity,  and  so  on  ; that  is,  the  motive  force  is  as  the 
velocity  ; but  the  momentum  impressed,  is  as  the  force  which 
produces  it,  by  prop.  2 ; and  therefore  the  momentum  is  as 
the  velocity  when  the  mass  is  the  same.  But  the  momentum 
was  found  to  be  as  the  mass  when  the  velocity  is  the  same. 

Consequently, 


GENERAL  LAWS  OF  MOTION. 


11& 

Consequently,  when  neither  are  the  same,  the  momentum  is 
in  the  compound  ratio  of  both  the  mass  and  velocity. 

PROPOSITION  IV. 

22.  In  Uniform  Motions,  the  Spaces  described  are  in  the  Com- 
pound Ratio  of  the  Velocities  and  the  Times  of  their  Descrip- 
tion. 

That  is,  s is  as  tv. 

For,  by  the  nature  of  uniform  motion,  the  greater  the  ve- 
locity. the  greater  is  the  space  described  in  any  one  and  the 
same  time  ; that  is,  the  space  is  as  the  velocity,  when  the 
times  are  equal  And  when  the  velocity  is  the  same,  the  space 
will  be  as  the  time  ; that  is,  in  a double  time  a double  space 
will  be  described  ; in  a triple  time,  a triple  space  ; and  so  on. 
Therefore  universally,  the  space  is  in  the  compound  ratio  of 
the  velocity  and  the  time  of  description. 

23.  Corol.  !.  In  uniform  motions,  the  time  is  as  the  space 
directly,  and  velocity  reciprocally  ; or  as  the  space  divided  by 
the  velocity.  And  when  the  velocity  is  the  same,  the  time  is 
as  the  space.  But  when  the  space  is  the  same,  the  time  is  re- 
ciprocally as  the  velocity. 

24.  Corol.  2 The  velocity  is  as  the  space  directly  and  the 
time  reciprocally  ; or  as  the  space  divided  by  the  time.  And 
when  the  time  is  the  same,  the  velocity  is  as  the  space.  But 
when  the  space  is  the  same,  the  velocity  is  reciprocally  as  the 
ti  me. 

Scholium. 

25.  In  uniform  motions  generated  by  momentary  impulse, 
let  b = any  body  or  quantity  of  matter  to  be  moved, 

f = force  of  impulse  acting  on  the  body  b , 
v = the  uniform  velocity  generated  in  6, 
m = the  momentum  generated  in  6, 
s = the  space  described  by  the  body  b, 
i = the  time  of  describing  the  space  s with  the  veloc.  v . 

Then  from  the  last  three  propositions  and  corollaries,  we 
have  these  three  general  proportions,  namely/  oc  m,  m a bv, 
and  s oc  tv  ; from  which  is  derived  the  following  table  of  the 
general  relations  of  those  six  quantities,  in  uniform  motions 
and  impulsive  or  percussive,  forces  : 

Vo*.  II.  ' 16 

6 


f <x  m 


1 14 


OF  MOTION,  FORCES,  &c. 


f ot  m at  bv  ot  — . 

J t 

m ot  /aha 
, f m ft  mt 

b a - a - a - a — . 

fl  tm 

s ot  tv  ot  - ot  — . 

o 6 

s f m 

v ot  - ot  7 a r. 

. * As  As 

2 a - a - a — . 

v J m 


By  mean?  of  which,  may  he  resolved  all  questions  relating 
to  uniform  motions,  and  the  effects  of  momentary  or  impulsive 
forces. 


PROPOSITION  V. 

26.  The  Momentum  generated  by  a Constant  and  Uniform  Foret 
acting  for  any  Time,  is  in  the  Compound  Ratio  of  the  Force 
and  Time  of  Acting. 

That  is,  m is  as  ft. 

For,  supposing  the  time  divided  into  very  small  parts, 
by  prop  2,  the  momentum  in  each  particle  of  time  is  the  same, 
and  therefore  the  whole  momentum  will  be  as  the  whole  time, 
or  sum  of  all  the  small  parts.  But  by  the  same  prop  the  mo- 
mentum for  each  small  time  is  also  as  the  motive  force.  Con- 
sequently the  whole  momentum  generated,  is  in  the  compound 
ratio  of  the  force  and  time  of  acting 

27.  Corol.  1 The  motion,  or  momentum,  lost  or  destroyed 
in  any  time,  is  also  in  the  compound  ratio  of  the  force  and 
time  For  whatever  momentum  any  force  generates  in  a 
given  time  ; the  same  momentum  will  an  equal  force  destroy 
in  the  same  or  equal  time  ; acting  in  a contrary  direction. 

And  the  same  is  true  of  the  increase  or  decrease  of  motion, 
by  forces  that  conspire  with,  or  oppose  the  motion  of  bodies 

28.  lorol  2 The  velocity  generated,  or  destroyed,  in  any 
time,  is  directly  as  the  force  and  time,  and  reciprocally  as  the 
body  or  mass  of  matter — For,  by  this  and  the  3d  prop,  the 
compaund  ratio  of  the  body  and  velocity,  is  as  that  of  the  force 
and  time  ; and  therefore  the  velocity  is  as  the  force  and  time 
divided  by  the  body  And  if  the  body  and  force  be  given,  or 
constant,  the  velocity  will  be  as  the  time. 


PROPOSITION 


GENERAL  LAWS  OF  MOTION. 


S15 


PROPOSITION  VI. 

29.  The  Spaces  passed  over  by  Bodies,  urged  by  any  Constant 
and  Uniform  Forces,  acting  during  any  Tiroes,  are  in  the  com- 
pound Ratio  of  the  Forces  and  Squares  of  the  Times  direct- 
ly, and  the  Body  or  Mass  reciprocally. 

Or,  the  Spaces  are  as  the  Squares  of  the  Times,  when  the  Force 
and  Body  are  given. 

That  is,  s is  as  or  as  t2  when  f and  b are  given.  For, 
b 

let  v denote  the  velocity  acquired  at  the  end  of  any  time  t, 
by  any  given  body  b,  when  it  has  passed  over  the  space  s. 
Then,  because  the  velocity  is  as  the  time,  by  the  last  corol. 
therefore  \ v is  the  velocity  at  A t,  or  at  the  middle  point  of 
the  time  ; and  as  the  increase  of  velocity  is  uniform,  the 
same  space  s will  be  described  in  the  same  time  t,  by  the 
velocity  A v,  uniformly  continued  from  beginning  to  end. 
But,  in  uniform  motions,  the  space  is  in  the  compound  ratio 
of  the  time  and  velocity  ; therefore  s is  as  A tv,  or  indeed  s = 

But,  by  the  last  corol.  the  velocity  v is  as  ^f-,oras 

the  force  and  time  directly,  and  as  the  body  reciprocally. 

f/2 

Therefore,  s,  or  \ tv,  is  as  that  is,  the  spaceisas  the  force 

and  square  of  the  time  directly,  and  as  the  body  reciprocally. 
Or  5 is  as  t2 , the  square  of  the  time  only,  when  b and  f are 
given. 

30  Corol  1 The  space  s is  also  as  tv,  or  in  (he  com- 
pound ratio  of  the  time  and  velocity  ; b and/ being  given. 
For,  s = \tv  is  the  space  actually  described  But  tv  is  the 
space  which  might  be  described  in  the  same  time  t.  with  the 
last  velocity  v,  if  it  were  uniformly  continued  for  the  same  or 
an  equal  time.  Therefore  the  space  s,  or  ±tv,  which  is  ac-* 
tually  described,  is  just  half  the  space  tv,  which  would  be  de- 
scrihed  with  the  last  or  greatest  velocity,  uniformly  continu- 
ed for  an  equal  time  t 

31.  Corol.  2.  The  space  s is  also  as  v2 , the  square  of  the 
velocity  ; because  the  velocity  v is  as  the  time  t. 


Scholium. 


32.  Propositions  3,  4,  5,  6,  give  theorems  for  resolving  all 
questions  relating  to  motions  uniformly  accelerated.  Thus, 

put 


116 


OF  MOTION,  FORCES,  &c 


put  b = any  body  or  quantity  of  matter, 
f = the  force  constantly  acting  on  it, 
t = the  time  of  its  acting, 
v — the  velocity  generated  in  the  time  t, 
s = the  space  described  in  that  time, 
m = the  momentum  at  the  end  of  the  time. 

Then,  from  these  fundamental  relations,  m oc  bv,  m te  ft, 
3 oc  tv,  and  v oc  *— , we  obtain  the  following  table  of  the 
general  relations  of  uniformly  accelerated  motions  : 

, - bs  fs  ftzv 

m oc  bv  on  ft  ce  — oc  — oc  


. m ft  mt 
b oc  - oc  cc  — 


f,i  pt3 

oc  — oc  ' 

s ms 


/m  bv  mv  ms  m3 

oc  — «c  K — OC  OC  — - oc 

t t S t2v  08 


ft 


v 06  r * ~b  * 


b * 


bft 


y/bfs  OC 

■J  bftv. 

m2 

m3 

* fs 

fi 

cc  — — 
Jtv 

* 15  ‘ 

m2 

bv 2 

bs 

btv 

oc 

s 

■ oc  • 

t 2 

a; 

S-  cc 

f2*t 

V b * 

m2 

fit  mt  ft*v  mro  ms 

S oc  tv  oc  — oc  -7-  oc  oc  — 0:  — oc 

b b m f bf 


bv 3 

~T 


m** 


- OC  - a: 
® / 


bv 

7 


bs 


bs  ms  m2 

00  - oc  y/-  OC  y/—r  K — , &C. 
m f ■ fv  bjv 


33.  And  from  these  proportions  those  quantities  are  to  be 
left  out  which  are  given,  or  which  are  proportional  to  each 
other.  Thus,  if  the  body  or  quantity  of  matter  be  always 
the  same,  then  the  space  described  is  as  the  force  and  square 
of  the  time.  And  if  the  body  be  proportional  to  the  force, 
as  all  bodies  are  in  respect  to  their  gravity  ; then  the  space 
described  is  as  the  square  of  the  time,  or  square  of  the  velo- 


city ; and  in  this  case,  if  f be  put 
force  ; then  will 


tv 


vS 

Ft'  oc  — . 


y , the  accelerating 

O 


v cc  - oc  Ft  et  y/rs 
t 


to c - oc  - ec  - 

V F v r 


THE 


GENERAL  LAWS  OF  MOTION. 


IK 


THE  COMPOSITION  AND  RESOLUTION  OF 
FORCES. 

34.  Composition  of  Forces,  is  the  uniting  of  two  or 
more  forces  into  one,  which  shall  have  the  same  effect ; or 
the  finding  of  one  force  that  shall  be  equal  to  several  others 
taken  together,  in  any  different  directions.  And  the  resolu- 
tion of  Forces,  is  the  finding  of  two  or  more  forces  which,, 
acting  in  any  different  directions,  shall  have  the  same  effect 
at  any  given  single  force. 

PROPOSITION  VII. 

35.  If  a Body  at  a be  urged  in  the  Directions  ab  and  ac,  by  any 
two  Similar  Forces , such  that  they  would  separately  cause  the 
Body  to  pass  over  the  Spaces  ab,  ac,  in  an  equal  Time  ; then 
if  both  Forces  act  together,  they  will  cause  the  Body  to  move  in 
the  same  Time,  through  ad  the  Diagonal  of  the  Parallelogram 
ABCD. 


Draw  cd  parallel  to  ab,  and  bd  paral- 
lel to  ac.  And  while  the  body  is  carried 
over  a b,  or  cd  by  the  force  in  that  direc-  . . . 

tion,  let  it  be  carried  over  bd: by  the  force  cj— 
in  that  direction  ; by  which  means  it  will 
he  found  at  d Now,  if  the  forces  be 
impulsive  or  momentary,  the  motions  will 
he  uniform,  and  the  spaces  described  will  be  as  the  times  of 
description  : 


tlieref.  a b or  cd  : ab  or  cd  : : time  in  a b : time  in  ab, 

and  bd  or  ac  : bd  or  ac  : : time  in  ac  : time  in  ac  ; 

but  the  time  in  Ab  : = time  in  ac,  and  the  time  in  ab  = 
time  in  ac  ; therefore  a b : bd  : : ab  : bd  by  equality  : hence 
the  point  d is  in  the  diagonal  ad. 

And  as  this  is  always  the  case  in  every  point  d,  d,  &c.  there- 
fore the  path  of  the  body  is  the  straight  line  Ada,  or  the  di- 
agonal of  the  parallelogram. 

But  if  the  similar  forces,  by  means  of  which  the  body  is 
moved  in  the  directions  ab,  ac,  be  uniformly  accelerating 
ones,  then  the  spaces  will  be  as  the  squares  of  the  times  ; in 
which  case,  call  the  time  in  bd  or  cd,  t,  and  the  time  in  ab  or 
ac,  t ; then 

it  will  be  a b or  cd  : ab  or  cd  : : t2  : t2, 

and  - bd  or  ac  : bd  or  ac  : : t2  : t3, 

theref,  by  equality,  Ab  : bd  : : ab  : bd  ; 
and  so  the  body  is  always  found  in  the  diagonal,  as  before. 

36.  Carol'. 


118 


OF  MOTION,  FORCES,  &c. 


36.  Corol.  1.  If  the  forces  be  not  similar,  by  which  the 
body  is  urged  in  the  directions  ab,  ac,  it  will  move  in  some 
curved  line,  depending  on  the  nature  of  the  forces. 

37.  Corol.  2.  Hence  it  appears  that  the  body  moves  over 
the  diagonal  ad,  by  the  compound  motion,  in  the  very  same 
time  that  it  would  move  over  the  side  ab  by  the  single  force 
impressed  in  that  direction,  or  that  it  would  move  over  the 
side  ac  by  the  force  impressed  in  that  direction. 

38  Corol.  3.  The  forces  in  the  directions,  ab,  ac,  ad.  are 
respectively  proportional  to  the  lines  ab,  ac,  ad,  and  in  these 
directions. 

39.  Corol.  4.  The  two  oblique  forces 
ab,  ac,  are  equivalent  to  the  single  di- 
rect force  ad,  which  may  be  compound- 
ed of  these  two,  by  drawing  the  diagonal 
of  the  parallelogram.  Or  they  are  equi- 
valent to  the  double  of  ae,  drawn  to  the 
middle  of  the  line  bo.  And  thus  any 
force  may  be  compounded  of  two  or  more 
other  forces  ; which  is  the  meaning  of  the  expression  compo- 
sition of forces. 

40.  Exam.  Suppose  it  were 
required  to  compound  the  three 
forces  ab,  ac,  ad  ; or  to  find  the 
direction  and  quantity  of  one  sin- 
gle force  which  shall  be  equi- 
valent to,  and  have  the  same 
effect,  as  if  a body  a were  acted 
on  by  three  forces  in  the  directions  ab,  ac,  ad,  and  propor- 
tional to  these  three  lines  First  reduce  the  two  ac,  ad.  to 
one  ae,  by  completing  the  parallelogram  adec.  Then  re- 
duce the  two  ae,  ab  to  one  af  by  the  parallelogram  aefb. 
So  shall  the  single  force  af  be  the  direction,  and  as  the  quan- 
tity. which  shall  of  itself  produce  the  same  effect,  as  if  all  the 
three  ab,  ac,  ad  acted  together. 

41.  Corol  5.  Hence  also  any  single 
direct  force  ad,  may  be  resolved  into 
two  oblique  forces,  whose  quantities 
and  directions  are  ab,  ac,  having  the 
same  effect,  by  describing  any  paralle- 
logram whose  diagonal  may  be  ad  : and 
this  is  called  the  resolution  of  forces. 

So  the  force  ad  may  be  resolved  into 
the  two  ab,  ac  by  the  parallelogram 


ABDC, 


GENERAL  LAWS  OF  MOTION. 


119 


abdc  ; or  into  the  two  ae,  af,  by  the  parallelogram  aedf  ; 
and  so  on,  for  any  other  two.  And  each  of  these  may  be  re- 
solved again  into  as  many  others  as  we  please. 

42.  Carol  6.  Hence  too  may  be 
found  the  effect  of  any  given  force,  in 
any  other  direction,  besides  that  of  the 
line  in  which  it  acts  ; as,  of  the  force 
ab  in  any  other  given  direction  cb  For 
draw  ad  perpendicular  to  cb  ; then  shall 
db  be  the  effect  of  the  force  ab  in  the  di- 
rection cb.  For  the  given  force  ab  is  equivalent  to  the  two 
ad,  db,  or  ae  ; of  which  the  former  ad,  or  eb,  being  perpen- 
dicular, does  not  alter  the  velocity  in  the  direction  cb  ; and 
therefore  db  is  the  whole  effect  of  ab  in  the  direction  cb. 
That  is,  a direct  force  expressed  by  the  line  db  acting  in  the 
direction  db,  will  produce  the  same  effect  or  motion  in  a body 
B,  in  that  direction,  as  the  oblique  force  expressed  by,  and 
acting  in,  the  direction  ab,  produces  in  tire  same  direction  cb. 
And  hence  any  given  force  ab,  is  to  its  effect  in  db,  as  ab  to 
db,  or  as  radius  to  the  cosine  of  the  angle  abd  of  inclination 
of  those  directions.  For  the  same  reason,  the  force  or  effect 
in  the  direction  ab,  is  to  the  force  or  effect  in  the  direction  ad 
or  eb,  as  ab  to  ad  ; or  as  radius  to  sine  of  the  same  angle  abd, 
•r  cosine  of  the  angle  dab  of  those  directions. 

43.  Carol.  7.  Hence  also,  if  the  two  given  forces,  to  be 
compounded,  act  in  the  same  line,  either  both  the  same  way, 
or  the  one  directly  opposite  to  the  other  ; then  their  joint  or 
compounded  force  will  act  in  the  same  line  also,  and  will  be 
equal  to  the  sum  of  the  two  when  they  act  the  same  way,  or 
to  the  difference  of  them  when  they  act  in  opposite  direc- 
tions ; and  the  compound  force,  whether  it  be  the  sum  or  dif- 
ference, will  always  act  in  the  direction  of  the  greater  of  the 
.two. 

PROPOSITION  vin. 

44.  If  Three  Forces  a,  b,  c,  acting  all  together  in  the  same  Plane, 
keep  one  another  in  Equilibria  ; they  will  be  proportional  to 
the  Three  Sides  de,  ec,  cd,  of  a Triangle,  which  are  drawn 
Parallel  to  the  Directions  of  the  Forces  ad,  db,  cd. 

Produce  a»,  bd,  and  draw  cf,  ce  parallel  to  them. 

Then 


120 


OF  MOTION,  FORCES,  &c. 


Then  the  force  in  cd  is  equivalent 
to  the  two  ad,  bd,  by  the  supposi- 
tion ; but  the  force  cd  is  also  equi- 
valent to  the  two  ed  and  ce  or  fd  ; 
therefore,  if  cd  represent  the  force 
c,  then  ed  will  represent  its  opposite 
force  a,  and  ce,  or  fd,  its  opposite 
force  b.  Consequently  the  three 
forces,  a,  b,  c,  are  proportional  to  de, 
ce,  cd,  the  three  lines  parallel  to  the 
directions  in  which  they  act. 

45.  Carol.  1.  Because  the  three  sides  cd,  ce,  de,  are  pro- 
portional to  the  sines  of  their  opposite  angles  e,  d,  c ; there- 
fore the  three  forces,  when  in  equilibrio,  are  proportional  to 
the  sines  of  the  angles  of  the  triangle  made  of  their  lines  of 
direction  ; namely,  each  force  proportional  to  the  sine  of  the 
angle  made  by  the  directions  of  the  other  two. 

46.  Corol.  2.  The  three  forces,  acting  against,  and  keep- 
ing one  another  in  equilibrio,  are  also  proportional  to  the  sides 
of  any  other  triangle  made  by  drawing  lines  either  perpendi- 
cular to  the  directions  of  the  forces,  or  forming  any  given 
angle  with  those  directions.  For  such  a triangle  is  always 
similar  to  the  former,  which  is  made  by  drawing  lines  parallel 
to  the  directions  ; and  therefore  their  sides  are  in  the  same 
proportion  to  one  another. 

47.  Corol.  3.  If  any  number  of  forces  be  kept  in  equilibrio 
by  their  actions  against  one  another  ; they  may  be  all  reduced 
to  two  equal  and  opposite  ones. — For,  by  cor  4,  prop.  7,  any 
two  of  the  forces  may  be  reduced  to  one  force  acting  in  the 
same  plane  ; then  this  last  force  and  another  may  likewise  be 
reduced  to  another  force  acting  in  their  plane  ; and  so  on.  till 
at  last  they  all  be  reduced  to  the  action  of  only  two  opposite 
forces  ; which  will  be  equal,  as  w'eli  as  opposite,  because  the 
whole  are  in  equilibrio  by  the  supposition. 

48.  Corol.  4.  If  one  of  the  forces, 
as  c,  be  a weight,  which  is  sustained 
by  two  strings  drawing  in  the  direc- 
tions da,  db  : then  the  force  or  ten- 
sion of  the  string  ad,  is  to  the  weight 
c,  or  tension  of  the  string  dc,  as  de 
to  dc  ; and  the  force  or  tension  of 
the  other  string  bd,  is  to  the  weight 
c,  or  tension  of  cd,  as  ce  to  cd. 


49.  Corol . 


COLLISION  OF  BODIES. 


121 


49.  Corol.  5.  If  three  forces  be  in  equilibrio  by  their  mutu- 
al actions  ; the  line  of  direction  of  each  force,  as  dc,  passes 
through  the  opposite  angle  c of  the  parallelogram  formed  by 
the  directions  of  the  other  two  forces. 

50.  Remark  These  properties,  in  this  proposition  and  its 
corollaries,  hold  true  of  all  similar  forces  whatever,  whether 
they  be  instantaneous  or  continual,  or  w'hether  they  act  by 
percussion,  drawing,  pushing,  pressing,  or  weighing  ; and  are 
of  the  utmost  importance  in  mechanics  and  the  doctrine  of 
forces. 


ON  THE  COLLISION  OF  BODIES. 


PROPOSITION  IX. 

51.  If  a Body  strike  or  act  Obliquely  on  a Plain  Surface , the 
Force  or  Energy  of  the  Stroke , or  Action,  is  as  the  Sine  of  the 
dingle  of  Incidence. 

Or,  the  Force  on  the  Surface  is  to  the  same  if  it  had  acted  Perpen- 
dicularly, as  the  Sine  of  Incidence  is  to  Radius. 


Let  ab  express  the  direction  and 
the  absolute  quantity  of  the  oblique 
force  on  the  plane  de  ; or  let  a given 
body  a,  moving  with  a certain  velo- 
city, impinge  on  the  plane  at  b ; 
then  its  force  will  be  to  the  action 
on  the  plane,  as  radius  to  the  sine  of 
the  angle  abd,  or  as  ab,  to  ad  or  bc,  drayring  ad  and  bc 
pendicular,  and  ac  parallel  to  de. 

For,  by  prob.  7,  the  force  ab  is  equivalent  to  the  two  forces 
ac,  cb  ; of  which  the  former  ac  does  not  act  on  the  plane,  be- 
cause it  is  parallel  to  it.  The  plane  is  therefore  only  acted  on 
by  the  direct  force  cb,  which  is  to  ab,  as  the  sine  of  the  an- 
gle bac,  or  abd,  to  radius. 


per- 


52.  Corol.  1.  If  a body  act  on  another,  in  any  direction,  and 
by  any  kind  of  force,  the  action  of  that  force  on  the  second 
body,  is  made  only  in  a direction  perpendicular  to  the  surface 
on  which  it  acts.  For  the  force  in  ab  acts  on  de  only  by  the 
force  cb,  and  in  that  direction. 


53.  Corol.  2.  If  the  plane  de  be  not  absolutely  fixed,  it 
will  move,  after  the  stroke,  in  the  direction  perpendicular  to 
its  surface.  For  it  is  in  that  direction  that  the  force  is  ex- 
erted. 

Vor,.  II, 


17 


PROPOSITION 


122 


OF  MOTION,  FORCES,  &c 


PROPOSITION  X. 

64.  If  one  Body  a,  strike  another  Body  b,  which  is  either  at  Res" 
or  moving  towards  the  Body  a,  or  moving  from  it,  but  with  a 
less  Velocity  than  that  of  a,  then  the  Momenta,  or  Quantities  of 
Motion,  of  the  two  Bodies,  estimated  in  any  one  Direction, 
will  be  the  very  same  after  the  Stroke  that  they  were  before  it. 

For,  because  action  and  reaction  are  always  equal,  and  isi 
contrary  directions,  whatever  momentum  the  one  body  gains 
one  way  by  the  stroke,  the  other  must  just  lose  as  much  in 
the  same  direction  ; and  therefore  the  quantity  of  motion  in 
that  direction,  resulting  from  the  motions  of  both  the  bodies 
remains  still  the  same  as  it  was  before  the  stroke. 

55.  Thus,  if  a with  a momentum 

of  10,  strike  b at  rest,  and  commu-  O O 

aicate  to  it  a momentum  of  4,  in  the  .A.  B 

direction  ab.  Then  a will  have  only 

a momentum  of  6 in  that  direction  ; which,  together  with  the 
momentum  of  b,  viz.  4,  make  up  still  the  same  momentum  be- 
tw’een  them  as  before,  namely,  10. 

56.  If  b were  in  motion  before  the  stroke  with  a momen- 
tum of  5,  in  the  same  direction,  and  receive  from  a an  addi- 
tional momentum  of  2.  Then  the  motion  of  a after  the  stroke 
will  be  8,  and  that  of  b,  7 ; which  between  them  make  15,  the 
same  as  10  and  5,  the  motions  before  the  stroke. 

57.  Lastly,  if  the  bodies  move  in  opposite  directions,  and 
meet  one  another,  namely,  a with  a motion  of  10,  and  b,  of 
5 ; and  a communicate  to  b a motion  of  6 in  the  direction  ab 
of  its  motion.  Then,  before  the  stroke,  the  ryhole  motion 
from  both,  in  the  direction  of  ab,  is  10  — 5 or  5.  But,  after 
the  stroke,  the  motion  of  a is  4 in  the  direction  ab,  and  the 
motion  of  b is  6 — 5 or  1 in  the  same  direction  ab  ; therefore 
the  sum  4 + 1,  or  5,  is  still  the  same  motion  from  both  as  it 
wras  before. 


PROPOSITION  XI. 

58.  The  Motion  of  Bodies  included  in  a Given  Space,  is  the 
same  with  regard  to  each  other,  whether  that  Space  be  at  Rest, 
or  move  uniformly  in  a Right  Line. 

For,  if  any  force  be  equally  impressed  both  on  the  body 
and  the  line  in  which  it  moves  this  will  cause  no  change  ist 

the 


COLLISION  OP  BODIES. 


123 

the  motion  of  the  body  along  the  right  line.  For  the  same 
reason,  the  motions  of  all  the  other  bodies,  in  their  several 
directions,  will  still  remain  the  same.  Consequently  their 
motions  among  themselves  will  continue  the  same,  whether 
'the  including  space  be  at  rest,  or  be  moved  uniformly  for- 
ward. And  therefore  their  mutual  actions  on  one  another, 
must  also  remain  the  same  in  both  cases. 


PROPOSITION  XII. 

59.  If  a Hard  and  Fixed  Plane  be  struck  by  either  a Soft  or  a 
Hard  Unelastic  Body , the  Body  will  continue  on  it.  But  if  the 
Plane  be  struck  by  a Perfectly  Elastic  Body , it  will  rebound 
from  it  again  with  the  same  Velocity  with  which  it  struck  the 
Plane. 

For,  since  the  parts  which  are  struck,  of  the  elastic  body* 
suddenly  yield  and  give  way  by  the  force  of  the  blow,  and 
as  suddenly  restore  themselves  again  with  a force  equal  to 
the  force  which  impressed  them,  by  the  definition  of  elastic 
bodies  ; the  intensity  of  the  action  of  that  restoring  force  on 
the  plane,  will  be  equal  to  the  force  or  momentum  with 
which  the  body  struck  the  plane.  And,  as  action  and  re- 
action are  equal  and  contrary,  the  plane  will  act  with  the 
same  force  On  the  body,  and  so  cause  it  to  rebound  or  move 
back  again  with  the  same  velocity  as  it  had  before  the  stroke. 

But  hard  or  soft  bodies,  being  devoid  of  elasticity,  by  the 
-definition,  having  no  restoring  force  to  throw  them  off  again, 
they  must  necessarily  adhere  to  the  plane  struck. 

60.  Corol.  1.  The  effect  of  the  blow  of  the  elastic  body, 
-on  the  plane,  is  double  to  that  of  the  unelastic  one,  the  velo 
eity  and  mass  being  equal  in  each 

For  the  force  of  the  blow  from  the  unelastic  body  is  as 
its  mass  and  velocity,  which  is  only  destroyed  by  the  resist- 
ance of  the  plane.  But  in  the  elastic  body,  that  force  is  not 
only  destroyed  and  sustained  by  the  plane  ; but  another  also 
equal  to  it  is  sustained  by  the  plane,  in  consequence  of  the 
restoring  force,  and  by  virtue  of  which  the  body  is  thrown 
back  again  .with  an  equal  velocity.  And  therefore  the  in- 
tensity of  the  blow  is  doubled. 

61.  Corol.  2.  Hence  unelastic  bodies  k^e,  by  their  col- 
lision, only  half  the  motion  lost  by  elasticpaodies  ;*  their  mass 
and  velocities  being  equal.— For  the  latter  communicate, 
double  the  motion  ef  the  former. 


PROPOSITION 


124 


OF  MOTION,  FORCES,  kc. 


PROPOSITION  XIII. 

4 

62.  If  an  Elastic  Body  a impinge  on  a Firm  Plane  de  at  the, 
Pvint  b,  it  will  rebound  from  it  in  an  Jingle  equal  to  that  in- 
which  it  struck  it ; or  the  Angle  of  Incidence  will  be  equal  to 
the  Angle  of  Reflexion;  namely , the  Angle  abd  equal  to  the 
Angle  eee.  • 

Let  ab  express  the  force  of 
the  body  a in  the  direction  ab  ; 
which  let  be  resolved  into  the 
two  ac,  cb,  parallel  and  per- 
pendicular to  the  plane. — 'i  ake 
be  and  cf  equal  to  ac,  and  draw 
BF.  Now  action  and  reaction  being  equal,  the  plane  will 
resist  the  direct  force  cb  by  another  bc  equal  to  it,  and  in  a 
contrary  direction  ; whereas  the  other  ac,  being  parallel  to 
the  plane,  is  not  acted  on  or  diminished  by  it,  but  still  con- 
tinues as  before.  The  body  is  therefore  reflected  from  the 
plane  by  two  forces  bc.  be,  perpendicular  and  parallel  to  the 
plane,  and  therefore  moves  in  the  diagonal  bf  by  composition. 
But,  because  ac  is  equal  to  be  or  cf,  and  that  ec  is  common, 
the  two  triangles  bca,  bcf  are  mutually  similar  and  equal  ; and 
consequently  the  angles  at  a and  f are  equal,  as  also  their 
equal  alternate  angles  abd,  fbe,  which  are  the  angles  of  inci- 
dence and  reflexion. 

PROPOSITION  XIV. 

63.  To  determine  the  Alotion  of  JVon-clastic  Bodies  when  they 
strike  each  other  Directly , or  in  the  same  Line  of  Direction. 

Let  the  non-elastic  body  b,  mov- 
ing with  the  velocity  v in  the  di-  0 

rection  b b,  and  the  body  b with  y C 

the  velocity  v,  strike  each  other. 

Then,  because  the  momentum  of  any  moving  body  is  as  the 
mass  into  the  velocity,  bv  — ii  is  the  momentum  of  the  body, 
b,  and  bv  = m the  momentum  of  the  body  6,  which  let  be 
the  less  powerful  of  the  two  motions.  Then,  by  prop.  10. 
the  bodies  will  both  move  together  as  oDe  mass  in  the  direc- 
tion bc  after  the  stroke,  whether  before  the  stroke  the  body 
b moved  towaids*^'  or  towards  b.  Now,  according  as  that 
motion  o I*b  was  4om  or  towards  b,  that  is  whether  the 
(notions  were  in  the  same  or  contrary  ways,  the  momentum 
after  the  stroke,  in  direction  bc,  will  be  the  sum  of  difterence 


COLLISION  OF  BODIES. 


125 


of  the  momentums  before  the  stroke  ; namely,  the  momen- 
tum in  direction  bc  will  be 


bv  -f  bv,  if  the  bodies  moved  the  same  way,  or 

bv bv,  if  they  moved  contrary  ways,  and 

bv  only,  if  the  body  b were  at  rest. 


Then  diride  each  momentum  by  the  common  mass  of  mat- 
ter b + b,  and  the  quotient  will  be  the  common  velocity  after 
the  stroke  in  the  direction  bc  ; namely,  the  common  velocity 
will  be,  in  the  first  case, 


BV  + bv 

u + b ’ 


in  the  2d 


bv— bv 
'b  -M  ’ 


and  in  the  3d 


BV 

B-j-£ 


64.  For  example,  if  the  bodies,  or  weights,  b and  b,  be  as 
5 to  3 and  their  velocities  v and  v,  as  6 to  4,  or  as  3 to  2,  be- 
fore the  stroke  ; then  15  and  6 will  be  as  their  momentums, 
and  8 the  sum  of  their  weights  ; consequently,  after  the  stroke, 
the  common  velocity  will  be  as 

15  + 6 or  2f  in  the  first  case, 

8 8 

*5  — 6 _ 9^or  ji  in  the  second,  and 
8 8 

.or  11  in  the  third. 

8 8 


PROPOSITION  XV. 

65.  If  two  Perfectly  Elastic  Bodies  impinge  on  one  another  : 
their  Relative  Velocity  will  be  the  same  both  Before  an(l  Af- 
ter the  Impulse  : that  is,  they  will  recede  from  each  other 
with  the  same  Velocity  with  which  they  approached  and  met. 

For  the  compressing  force  is  as  the  intensity  of  the  stroke  ; 
which,  in  given  bodies,  is  as  the  relative  velocity  with  which 
they  meet  or  strike  But  perfectly  elastic  bodies  restore 
themselves  to  their  former  figure,  by  the  same  force  by  which 
they  were  compressed  ; that  is,  the  restoring  force  is  equal 
to  the  compressing  force,  or  to  the  force  with  which  the 
bodies  approach  each  other  before  the  impulse.  But  the 
bodies  are  impelled  from  each  other  by  this  restoring  force  ; 
and  therefore  this  force,  acting  on  the  same  bodies,  will  pro- 
duce a relative  velocity  equal  to  that  which  they  had  before  : 
or  it  will  make  the  bodies  recede  from  each  other  with  the 

same 


QF  MOTION-,  FORCES,  &c. 


3 26 

same  velocity  with  which  they  before  approached,  or  so  as  to 
be  equally  distant  from  one  another  at  equal  times  before  and 
after  the  impact. 

66.  Remark.  It  is  not  meant  by  this  proposition,  that  each 
body  will  have  the  same  velocity  after  (he  impulse  as  it  had 
before  ; for  that  will  he  varied  according  to  the  relation  of 
the  masses  of  the  two  bodies  ; but  that  the  velocity  of  the 
one  will  be,  after  the  stroke,  as  much  increased  as  that  of 
the  other  is  decreased,  in  one  and  the  same  direction  So,  if 
the  elastic  body  b move  with  a velocity  v,  and  overtake  the 
elastic  body  b moving  the  same  way  with  the  velocity  v ; then 
their  relative  velocity,  or  that  with  which  they  strike,  is 
v — v,  and  it  is  with  this  same  velocity  that  they  separate 
from  each  other  after  the  stroke.  But  if  they  meet  each 
other,  or  the  body  b move  contrary  to  the  body  b ; then  they 
meet  and  strike  with  the  velocity  v + v,  and  it  is  with  the 
same  velocity  that  they  separate  and  recede  from  each  other 
after  the  stroke.  But  whether  they  move  forward  or  back- 
ward after  the  impulse,  and  with  what  particular  velocities, 
are  circumstances  that  depend  on  the  various  masses  and  ve- 
locities of  the  bodies  before  the  stroke,  and  which  make  the 
subject  of  the  next  proposition. 

PROPOSITION  XVI. 


67.  To  determine  (he  Motions  of  Elastic  Bodies  after  Striking 
each  other  directly. 

Let  the  elastic  bod}'  b move  in  „ 

the  direction  bo,  with  the  velocity  J C 

v ; s.nd  let  the  velocity  of  the  other 

body  b be  v in  the  same  line  ; which  latter  velocity  v will  be 
positive  if  b move  the  same  way  as  e,  but  negative  if  b move 
in  the  opposite  direction  to  b.  Then  their  relative  velocity 
in  the  direction  bc  is  v — v ; also  the  momenta  before  the 
stroke  are  bv  and  bv,  the  sum  of  which  is  bv  + bv  in  the 
direction  bc. 

Again,  put  x for  the  velocity  of  b,  and  y for  that  of  b, 
in  the  same  direction  bc,  al'thr  the  stroke  ; then  their  rela- 
tive velocity  is  y — x,  and  the  sum. of  their  momenta  e x-fby 
in  the  same  direction. 

But  the  momenta  before  and  after  the  collision,  estimated 
in  the  same  direction,  are  equal,  by  prop.  10,  as  also  the 
relative  velocities,  by  the  last  prop.  Whence  arise  these  two 
equations  : 

viz. 


COLLISION  OF  BODIES, 


-re? 


Tiz  bv  -f-  bv  — B.r  + by, 
and  v — v = y — x ; 

;he  resolution  of  which  equations  gives 

(b  — 6)  +26v  . . r 

x — - — - , the  velocity  of  b, 

— (b  - b)  V 4-  2b  V , ..  ~ , 

y =— ■ ^ , the  velocity  of  6, 

both  in  the  direction  bc,  when  v and  v are  both  positive,  or 
the  bodies  both  moved  towards  c before  the  collision.  But  if 
v be  negative,  or  the  body  6 moved  in  the  contrary  direction 
before  collision,  or  towards  b ; then,  changing  the  sign  of  v, 
the  same  theorems  become 


(b  b)  v , the  velocity  of  e, 


y = 


b + b 
_ (b  — b)  v -f  2b 


« | J 


, the  veloc.  of  b,  in  the  direction  bc. 


And  if  b were  at  rest  before  the  impact,  making  its  velocity 
p = 0,  the  same  theorems  give 


x =B  7~t  v,  and  y ==  — — v,  the  velocities  in  this  case. 

B + 0 B ~f-  6 

And  in  this  case,  if  the  two  bodies  b and  b be  equal  to 

each  other  ; then  b — b = 0,  and — = l • which 

b + 6 2b 

give  x = 0,  and  y ~ v ; that  is  the  body  b will  stand  still, 
and  the  other  body  b will  move  on  with  the  whole  velocity  of 
the  former  ; a thing  which  we  sometimes  see  happen  in  play- 
ing at  billiards  ; and  which  would  happen  much  oftener  if  the 
balls  were  perfectly  elastic. 


PROPOSITION  XV IP. 

68.  If  Bodies  strike  one  another  Obliquely,  it  is  proposed  to  de i 
termine  their  Motions  after  the  Stroke, 

Let  the  two  bodies  b,  b, 
move  in  the  oblique  directions 
ba,  6a,  and  strike  each  other 
at  a,  with  velocities  which  are 
in  proportion  to  the  lines  ba, 

Aa  ; to  find  their  motions  after 
the  impact.  Let  cah  repre- 
sent the  plane  in  which  the 
bodies  touch  in  the  point  of 
concourse  ; to  which  draw  the  perpendiculars  bc,  6d,  and 
complete  the  rectangles  ce,  df.  Then  the  motion  in  ba  is  re- 
solved 

i 


128 


OF  MOTION,  FORCES,  &c. 


solved  into  the  two  bc,  ca  ; and  the  motion  in  6a  is  resolvet 
into  the  two  6d,  da  ; of  which  the  antecedents  bc,  6d,  are  the 
velocities  with  which  they  directly  meet,  and  the  consequents 
ca,  da,  are  parallel  ; therefore  by  these  the  bodies  do  not  im- 
pinge on  each  other,  and  consequently  the  motions,  according 
to  these  directions,  will  not  be  changed  by  the  impulse  ; so 
that  the  velocities  with  which  the  bodies  meet,  are  as  ec  and 
6d,  or  their  equals  ea  and  fa.  The  motions  therefore  of  the 
bodies  b,  6,  directly  striking  each  other  with  the  velocities  ea, 
fa,  will  be  determined  by  prop.  1G  or  14,  according  as  the 
bodies  are  elastic  or  non-elastic  ; which  being  done,  let  ag  be 
the  velocity,  so  determined,  of  one  of  them,  as  b ; and  since 
there  remains  also  in  the  body  a force  of  moving  in  the  direc- 
tion parallel  to  be,  with  a velocity  as  be,  make  ah  equal  to  be, 
and  complete  the  rectangle  gh  :*  then  the  two  motions  in  ah 
and  ag,  or  hi,  are  compounded  into  the  diagonal  ai,  which 
therefore  will  be  the  path  and  velocity  of  the  body  b after  the 
stroke.  And  after  the  same  manner  is  the  motion  of  the  other 
body  6 determined  after  the  impact. 

If  the  elasticity  of  the  bodies  be  imperfect  in  any  given  de- 
gree, then  the  quantity  of  the  corresponding  lines  must  be  di- 
minished in  the  same  proportion. 

THE  LAWS  OF  GRAVITY  ; THE  DESCENT  OF  HEAVY 
BODIES  ; AND  THE  MOTION  OF  PROJECTILES  IN 
FREE  SPACE. 

PROPOSITION  xvm. 

69.  Jill  the  properties  of  Motion  delivered  in  Proposition  VI,  it: 
Corollaries  and  Scholium,  for  Constant  Forces,  are  true  in  the 
Motions  of  Bodies  freely  descending  by  their  orcn  Gravity  ; 
namely,  that  the  velocities  are  as  the  Times,  and  the  Spaces  as 
the  Squares  of  the  Times,  or  as  the  Squares  of  the  Veloci- 
ties. 

Fob,  since  the  force  of  gravity  is  uniform,  and  constantly 
the  same,  at  all  places  near  the  earth’s  surface,  or  at  nearly 
the  same  distance  from  the  centre  of  the  earth  ; and  since  this 
is  the  force  by  which  bodies  descend  to  the  surface  ; they 
therefore  descend  by  a force  which  acts  constantly  and  equally  : 
consequently  all  the  motions  freely  produced  by  gravity,  are 
as  above  specified,  by  that  proposition,  &c. 

SCHOLIUM. 

TO.  Now  it  has  been  found,  by  numberless  experiments 

tha* 


OF  GRAVITY. 


128 


that  gravity  is  a force  of  such  a nature,  that  all  bodies,  whether 
light  or  heavy,  fall  perpendicularly  through  equal  spaces  in 
the  same  time,  abstracting  from  the  resistance  of  the  air  ; as 
lead  or  gold  and  a feather,  which  in  an  exhausted  receiver 
fall  from  the  top  to  the  bottom  in  the  same  time.  It  is  also 
found  that  the  velocities  acquired  by  descending,  are  in  the 
exact  proportion  of  the  times  of  descent  : and  further,  that 
the  spaces  descended  are  proportional  to  the  squares  of  the 
times,  and  therefore  to  the  squares  of  the  velocities.  Hence 
then  it  follows,  that  the  weights  of  gravities,  of  bodies  near 
the  surface  of  the  earth,  are  proportional  to  the  quantities  of 
matter  contained  in  them  ; and  that  the  spaces,  times,  and 
velocities,  generated  by  gravity,  have  the  relations  contained 
in  the  three  general  proportions  before  laid  down.  Further, 
as  it  is  found,  by  accurate  experiments,  that  a body  in  the 
latitude  of  London,  falls  nearly  16tl  feet  in  the  first  second 
of  time,  and  consequently  that  at  the  end  of  that  time  it  has 
acquired  a velocity  double,  or  of  32}  feet  by  corol.  1,  prop.  6 ; 
therefore  if  g denote  16tl  feet,  the  space  fallen  through  in 
one  second  of  time,  or  2g  the  velocity  generated  in  that  time  ; 
then,  because  the  velocities  are  directly  proportional  to  the 
times,  and  the  spaces  to  the  squares  of  the  times  ; therefore 
it  will  be, 

as  l"  : t'  : : 2 g : 2gt  = v the  velocity, 
and  l3'  : t2  : : g : gl2  = s the  space. 

So  that,  for  the  descents  of  gravity,  we  have  these  genera! 
equations,  namely, 


s = gt2 


2s 


v = 2gt=  T = 2 sg. 


Hence,  because  the  times  are  as  the  velocities,  and  the 
spaces  as  the  squares  of  either,  therefore, 

if  the  times  be  as  the  numbs.  1,  2,  3,  4,  5,  &c. 

the  velocities  will  also  be  as  1,  2,  3,  4,  5,  &c. 

and  the  spaces  as  their  squares  1,  4,  9,  16,  25,  &c. 
and  the  space  of  each  time  as  1,3,  5,  7,  9,  &c. 

namely,  as  the  series  of  the  odd  numbers,  which  are  the 
differences  of  the  squares  denoting  the  w’hole  spaces.  So 
that  if  the  first  series  of  hatural  numbers  be  seconds  of  time, 
Vor.  II.  18  namely. 


130 


OF  MOTION,  FORCES,  &c. 


namely,  the  times  in  seconds 

1", 

2", 

3", 

. 4", 

kc. 

the  velocities  in  feet  will  be 

32*, 

64i, 

961, 

12P|, 

kc. 

the  spaces  in  the  whole  times 

16  A, 

64X, 

1441. 

2 

kc. 

and  the  space  for  each  second 

16tV, 

481, 

8(*t  2 > 

1 12-7- 
J U12> 

kc. 

71.  These  relations,  of  the  times,  veloci- 
ties, and  spaces,  may  be  aptly  represented 
by  certain  lines  and  geometrical  figures. 

Thus,  if  the  line  ab  denote  the  time  of  any 
body’s  descent,  and  bc,  at  right  angles  to  it, 
the  velocity  gained  at  the  end  of  that  time  ; 
by  joining  ac,  and  dividing  the  time  ab  into 
any  number  of  parts  at  the  points  a,  b,  c ; 
then  shall  ad,  be , cf,  parallel  to  bc,  be  the  velocities  at  the 
points  of  time  a,  b,  c,  or  at  the  ends  of  the  times,  a a,  a b, 
ac  ; because  these  latter  lines,  by  similar  triangles  are  pro- 
portional to  the  former  ad,  be,  cf,  and  the  times  are  propor- 
tional to  the  velocities.  Also,  the  area  of  the  triangle  abc 
will  represent  the  space  descended  through  by  the  force  of 
gravity  in  the  time  ab.  in  which  it  generates  the  velocity  bc  ; 
because  that  area  is  equal  to  IabXbc.  and  the  space  descend- 
ed is  s = \tv,  or  half  the  product  of  the  time  and  the  last 
velocity.  And,  for  the  same  reason,  the  less  triangles  a ad, 
Abe,  a cf,  will  represent  the  several  spaces  described  in  the 
corresponding  times  a a,  Ab,  ac,  and  velocities  ad,  be,  cf ; 
those  triangles  or  spaces  being  also  as  the  squares  of  their 
like  sides  a a,  Ab,  ac,  %vhich  represent  the  times,  or  of  ad,  be, 
cf,  which  represent  the  velocities. 


A 

a 

b 

c 


d 


B 


1 


P abed 


. 72.  But  as  areas  are  rather  unnatural 
representations  of  the  spaces  passed  over 
by  a body  in  motion,  which  are  lines,  the 
relations  may  better  be  represented  by 
the  abscisses  and  ordinates  cf  a parabola. 

Thus,  if  pq  be  a parabola,  pr  its  axis, 
and  rq  its  ordinate  ; and  pa,  p b,  pc.'&c. 
parallel  to  rq,  represent  the  times  from 
the  beginning,  or  the  velocities,  then  ae,  bf  cg,*k c.  jfarallel 
to  the  axis  pr,  will  represent  the  spaces  described  by  a fall- 
ing body  in  those  times  ; for,  in  a parabola,  the  abscisses  p h, 
pi,  p k,  kc.  or  ae,  bf,  eg,  &c.  which  are  the  spaces  described, 
are  as  the  squares  of  thS  ordinates  he.  if,  kg,  kc.  or  ra,  f b, 
pc,  kc.  which  represent  the  times  or  velocities. 


73.  And  because  the  laws  for  the  destruction  of  motion, 

are 


OF  GRAVITY. 


131 


are  the  same  as  those  for  the  generation  of  it,  by  equal  forces, 
but  acting  in  a contrary  direction  ; therefore, 

1st,  A body  thrown  directly  upward,  with  any  velocity  will 
lose  equal  velocities  in  equal  times. 

2 d,  if  a body  be  projected  upward,  with  the  velocity  it 
acquired  in  any  time  by  descending  freely,  it  will  lose  all  its 
velocity  in  an  equal  time,  and  will  ascend  just  to  the  same 
height  from  which  it  fell,  and  will  describe  equal  spaces  in 
equal,  times,  in  rising  and  falling,  but  in  an  inverse  order  ; 
and  it  will  have  equal  velocities  at  any  one  and  the  same  point 
of  the  line  described,  both  in  ascending  and  descending. 

3d,  If  bodies  he  projected  upward,  with  any  velocities,  the 
height  ascended  to.  will  be  as  the  squares  of  those  velocities, 
or  as  the  squares  of  the  times  of  ascending,  till  they  lose  all 
their  velocities. 

74.  l'o  illustrate  now  the  rules  for  the  natural  descent  of 
bodies  by  a few  examples;  let  it  be  required, 

1st,  To  find  the  space  descended  by  a body  in  7 seconds 
of  time,  and  the  velocity  acquired. 

Ans.  788^2  space  ; and  225£  velocity. 

2d,  To  find  the  time  of  generating  a velocity  of  100  feet 
per  second,  and  the  whole  space  descended. 

Ans.  3"t2J3  time  ; 155T8^  space. 

3d,  To  find  the  time  of  descending  400  feet,  and  the  velo- 
city at  the  end  of  that  time 

Ans.  4"i-f  time  ; and  160|^  velocity 


PROPOSITION  XIX. 

75.  If  a Body  be  projected  in  Free  Space  either  Parallel  to  the 
Horizon,  or  in  an  Oblique  Direction,  by  the  Force  of  Gun- 
Powder,  or  any  other  Impulse;  it  will  by  this  Motion , in 
Conjunction  with  the  Action  of  Gravity  describe  the  Curve 
Line  of  a Parabola. 


Let  the  body  be  projected-  from  the  point  a,  in  the  di- 
rection ad,  with  any  uniform  velocity  : then,  in  any  equal 

portions 


432 


OF  MOTION,  FORCES,  &c. 


portions  of  time,  it  would  by  prop.  4,  describe  the  equal 
spaces  ab,  bc,  cd,  &c.  in  the  line  ad,  if  it  were  not  drawn 
continually  down  below  that  line  by  the  action  of  gravity. 
Draw  be,  cf,  dg,  &c.  in  the  direction  of  gravity,  or  perpen- 
dicular to  the  horizon,  and  equal  to  the  spaces  through  which 
the  body  would  descend  by  its  gravity  in  the  same  time  in 
which  it  would  uniformly  pass  over  the  corresponding  spa<  es 
ab,  ac,  ad,  &c.  by  the  projectile  motion.  Then,  since  by 
these  two  motions  the  body  is  carried  over  the  space  ab,  in 
the  same  time  as  over  the  space  be,  and  the  space  ac 
in  the  same  time  as  the  space  cf,  and  the  space  ad  in  the 
same  time  as  the  space  dc,  &c.  ; therefore,  by  the  compo- 
sition of  motions,  at  the  end  of  those  times,  the  body  will  be 
found  respectively  in  the  points  e,  f,  g,  &.c.  ; and  consequent- 
ly the  real  path  of  the  projectile  will  be  the  curve  line  aefg, 
&c.  But  the  spaces  ab,  ac,  ad,  &c.  described  by  uniform  mo- 
tion, are  as  the  times  of  description  ; and  the  spaces  be,  cf,  dg, 
&c.  described  in  the  same  times  by  the  accelerating  force  of 
gravity,  are  as  the  squares  of  the  times  ; consequently  the 
perpendicular  descents  are  as  the  squares  of  the  spaces  in  ad, 
that  is  be,  cf,  dg,  &c.  are  respectively  proportional  to  ab2, 
ac2,  ad2,  &c.  ; which  is  the  property  of  the  parabola  by 
theor.  8,  Con.  beet.  Therefore  the  path  of  the  projectile  is 
the  parabolic  line  aefc,  &c.  to  which  ad  is  a tangent  at  the 
point  a. 

76.  Corol.  1.  The  horizontal  velocity  of  a projectile,  is 
always  the  same  constant  quantity,  in  every  point  of  the 
curve  ; because  the  horizontal  motion  is  in  a constant  ratio 
to  the  motion  in  ad,  which  is  the  uniform  projectile  motion. 
And  the  projectile  velocity  is  in  proportion  to  the  constant 
horizontal  velocity,  as  radius  to  the  cosine  of  the  angle  dam, 
or  angle  of  elevation  or  depression  of  the  piece  above  or  be- 
low the  horizontal  line  aii. 

77.  Corol.  2.  The  velocity  of  the  projectile  in  the  direction 
of  the  curve,  or  of  its  tangent  at  any  point  a is  as  the  secant 
of  its  angle  bai  of  direction  above  the  horizon  For  the 
motion  in  the  horizontal  direction  ai  is  constant,  and  ai  is  to 
ab,  as  radius  to  the  secent  of  the  angle  a ; therefore  the  mo- 
tion at  a,  in  ab,  is  every  where  as  the  secant  of  the  angle  a. 

78.  Corol.  3.  The  velocity  in  the  direction  dg  of  gravity, 
or  perpendicular  to  the  horizon,  at  any  point  g of  the  curve, 
is  to  the  first  uniform  projectile  velocity  at  a,  or  point  of 
contact  of  a tangent,  as  2gd  is  to  ad.  For,  the  times  in  ad 
and  dg  being  equal,  and  the  velocity  acquired  by  freely  de- 
scending 


PROJECTILES. 


1 33 


scending  through  dg,  being  such  as  would  carry  the  body  uni- 
formly over  twice  dg  in  an  equal  time,  and  the  spaces  describ- 
ed with  uniform  motions  being  as  the  velocities,  therefore  the 
space  ad  is  to  the  space  2dg,  as  the  projectile  velocity  at  a, 
to  the  perpendicular  velocity  at  g. 


PROPOSITION  XX. 


79.  The  Velocity  in  the  Direction  of  the  Curve,  at  any  Point  of 
it,  as  a,  is  equal  to  that  which  is  generated  by  Gravity  in 
freely  descending  through  a Space  which  is  equal  to  One- 
Fourth  of  the  Parameter  of  the  diameter  of  the  Parabola 
at  that  Point. 


Let  pa  or  ab  be  the  height 
due  to  the  velocity  of  the  projec- 
tile at  any  point  a,  in  the  direction 
of  the  curve  or  tangent  ac,  or 
the  velocity  acquired  by  fdlling 
through  that  height  ; and  com- 
plete the  parallelogram  acdb. 

Then  is  cd  = ab  or  ap,  the 
height  due  to  the  velocity  in  the  curve  at  a and  cois  also  the 
height  due  to  the  perpendicular  velocity  at  d,  which  must 
be  equal  to  the  former  ; but  by  the  last  corol  the  velocity  at 
a is  to  the  perpendicular  velocity  at  d,  as  ac  to  2cd  ; and 
as  these  velocities  are  equal,  therefore  ac  or  bd  is  equal  to 
2cd,  or  2ab  ; and  hence  ab  or  ap  is  equal  to  |bd,  or  i of  the 
parameter  of  the  diameter  ab,  by  corol.  to  theor.  13  of  the 
^Parabola. 


II  H H H K 


80.  Corol.  1 Hence,  and  from  cor.  2, 
theor.  13  of  the  parabola,  it  appears  that, 
if  from  the  directrix  of  the  parabola 
which  is  the  path  of  the  projectile,  seve- 
ral lines  he  be  drawn  perpendicular  to 
the  directrix,  or  parallel  to  the  axis  ; then 
the  velocity  of  the  projectile  in  the  direction  of  the  curve,  at 
any  point  e,  is  always  equal  to  the  velocity  acquired  by  a body 
falling  freely  through  the  perpendicular  line  he. 

81.  Corol.  2 If  a body,  after  falling  through  the  height 
pa  (last  fig.  but  one),  which  is  equal  to  ab,  and  when  it 
arrives  at  a,  have  its  course  changed,  by  reflection  from  an 
elastic  plane  ai,  or  otherwise,  into  any  direction  ac,  without 
altering  the  velocity  ; and  if  ac  be  taken  = 2ap  or  2ab, 

and 


134 


OF  MOTION,  FORCES,  &c. 


and  the  parallelogram  be  completed  ; then  the  body  will  de- 
scribe the  parabola  passing  through  the  point  n. 

82.  Corol  3.  Because  ac  = 2ab  or  2cd  or  2ap,  then  tore 
ac2  = 2ap  X 2cd  or  ap  . 4cd  ; and  because  all  the  perpen- 
diculars ef,  cd,  gh,  are  as  ae2,  ac2,  ag2  ; therefore  also 

ap  . 4ef  = ae2,  and  ap  . 4gh  = ag2,  &c.  ; and  because  the 

rectangle  of  the  extremes  is  equal  to  the  rectangle  of  the 
means  of  four  proportionals,  therefore  always 
it  is  ap  : ae  : : ae  4ef, 

and  ap  : ac  : : ac  : 4cd, 

and  ap  : ag  : : ag  : 4gh, 

and  so  on. 

PROPOSITION  XXI. 

83.  Having  given  the  Direction , and  the  Impetus,  or  Altitude 
due  to  the  First  Velocity  of  a Projectile  ; to  determine  the 
Greatest  Height  to  which  it  will  rise,  and  ike  Random  or 
Horizontal  Range. 

Let  ap  he  the  height  due  to  the 
projectile  velocity  at  a,  ag  the  di- 
rection, and  ah  the  horizon.  On 
ag  let  fall  the  perpendicular  pq, 
and  on  APthe  perpendicular  or  ; so 
shall  ar  be  equal  to  the  greatest  al- 
titude cv,and  4qr  equal  to  the  hori- 
zontal range  ah.  Or,  having  drawn 
pq  perp.  to  ag,  take  ag  = 4aq,  and  draw  gh  perp.  to  ah 
then  ah  is  the  range. 

For,  by  the  last  corollary,  ap  : >ag  : : ag  : 4gh  ; 

and,  by  similar  triangles,  ap  : ag  : : aq  : gh, 

or  - - - - ap  : ag  : : 4aq  : 4gh  ; 

therefore  ag  = 4aq  ; and,  by  similar  triangles,  ah  = 4qr. 

Also,  if  v be  the  vertex  of  the  parabola,  then  ab  or  Iag 
= 2aq,  or  aq.  = qb  ; consequently  ar  = bv,  which  is  =*=  cv 
bj'  the  property  of  the  parabola. 

84.  Corol.  1.  Because  the  angle 
q is  a right  angle,  which  is  the  angle 
in  a semicircle,  therefore  if,  on  ap 
as  a diameter,  a semicircle  be  de- 
scribed, it  will  pass  through  the 
point  q. 

85.  Corol.  2.  If  the  horizontal  range  O 

and  the  projectile  velocity  be  given, 

the  1 

ADC 


PROJECTILES. 


135 


the  direction  of  the  piece  so  as  to  hit  the  object  h,  will  he 
thus  easily  found  : Take  ad  = \ ah,  draw  dq  perpendicular 
to  ah,  meeting  the  semicircle,  described  on  the  diameter  ap, 
in  q and  q ; then  Aq  or  Aq  will  be  the  direction  of  the  piece. 
And  hence  it  appears,  that  there  are  two  directions  ab,  Ab, 
which,  with  the  same  projectile  velocity,  give  the  very  same 
horizontal  range  ah.  And  these  two  directions  make  equal 
angles  qAD,  qAP  with  ah  and  ap,  because  the  arc  pq  = the  arc 
Aq. 

86.  Corol.  3.  Or,  if  the  range  ah,  and  direction  ab.  be  giv- 
en ; to  find  the  altitude  and  velocity  or  impetus.  Take  ad  = 
Aah,  and  erect  the  perpendicular  oq.  meeting  ab  in  q ; so 
shall  Dq  be  equal  to  the  greatest  altitude  cv.  Also,  erect  ap 
perpendicular  to  ah,  and  qp  to  Aq  ; so  shall  ap  be  the  height 
due  to  the  velocity. 

87  Carol.  4.  When  the  body  is  projected  with  the  same 
velocity,  but  in  different  directions  : the  horizontal  ranges 
ah  will  be  as  the  sines  of  double  the  angles  of  elevation. — 
Or,  which  is  the  same,  as  the  rectangle  of  the  sine  and  co- 
sine of  elevation.  For  ad  or  Rq,  which  is  {ah,  is  the  sine 
of  the  arc  Aq,  which  measures  double  the  angle  qAD  of  eleva- 
tion. * 

And  when  the  direction  is  the  same,  but  the  velocities  diffe- 
rent ; the  horizontal  ranges  are  as  the  square  of  the  velocities, 
or  as  the  height  ap,  which  is  as  the  square  of  the  velocity  ; 
for  the  sine  ad  or  uq  or  ^ah  is  as  the  radius  or  as  the  diame- 
ter ap. 

Therefore,  when  both  are  different,  the  ranges  are  in  the 
compound  ratio  of  the  squares  of  the  velocities,  and  the  sines 
ef  double  the  angles  of  elevation. 

88.  Corol.  5.  The  greatest  range  is  when  the  angle  of  ele- 
vation is  45°,  or  half  a right  angle  ; for  the  double  of  45  is 
90,  which  has  the  greatest  sine.  Or  the  radius  os,  which  is 
| of  the  range,  is  the  greatest  sine. 

And  hence  the  greatest  range,  or  that  at  an  elevation  of  45°, 
is  just  double  the  altitude  ap  which  is  due  to  the  velocity,  or 
equal  to  4vc.  Consequently,  in  that  case,  c is  the  focus  of  the 
parabola,  and  ah  its  parameter.  Also,  the  ranges  are  equal, 
at  angles  equally  above  and  below  45°. 

89.  Corol.  6.  When  the  elevation  is  15°,  the  double  of 
which,  or  30°,  has  its  sine  equal  to  half  the  radius  ; conse- 
quently then  its  range  will  be  equal  to  ap,  or  half  the  greatest 
range  at  the  elevation  of  45°  ; that  is.  the  range  at  15°,  is 
equal  to  the  impetus  or  height  due  to  the  projectile  velocity. 

90.  Corol.  7. 


136 


OF  MOTION,  FORCES,  &c. 


90.  Corol.  7.  The  greatest  altitude  cv,  being  equal  to  ar, 
is  as  the  versed  sine  of  double  the  angle  of  elevatioe,  and 
also  as  ap  or  the  square  of  the  velocity.  Or  as  the  squ.  * 
of  the  sine  of  elevation,  and  the  square  of  the  velocity  ; for 
the  square  of  the  sine  is  as  the  versed  sine  of  the  double  an- 
gle. 

91.  Corol.  8.  The  time  of  flight  of  the  projectile,  which  is 
equal  to  the  time  of  a body  tailing  freely  through  gh  or 
4cv,  four  times  the  altitude,  is  therefore  as  the  square  root 
of  the  altitude,  or  as  the  projectile  velocity  and  sine  of  the  ele- 
vation. 


92.  From  the  last  proposition,  and  its  corollaries,  may  be 
deduced  the  following  set  of  theorems,  for  finding  all  the 
circumstances  of  projectiles  on  horizontal  planes,  having  anv 
two  of  them  given.  Thus,  let  s,  c,  t denote  the  sine,  cosine, 
and  tangent  of  elevation  ; s,  v the  sine  and  versed  sine  of 
the  double  elevation  ; r the  horizontal  range  ; t the  time  c 
flight ; v the  projectile  velocity  ; h the  greatest  height  of  the 
projectile  g = 16^2  feet,  and  a the  impetus,  or  the  altitud 


And  from  any  of  these,  the  angle  of  direction  may  be 
found.  Also,  in  these  theorems,  g may,  in  many  cases,  be 
taken  .=  16,  without  the  small  fractiou  TV , which  will  be  near 
enough  for  common  use. 


93.  To  determine  the  Range  on  an  Oblique  Plane;  having 
given  the  Impetus  or  Velocity,  and  the  Angle  oj  Direction. 


SCHOLIUM. 


due  to  the  velocity  v.  Then, 

sv 

r = 2as  = 4asc  = 


PROPOSITION  xm 


Let  ae  be  the  oblique  plane,  at  a given  angle,  either 
above  or  below  the  horizontal  plane  ah  ; ag  the  direction 

of 


PROJECTILES. 


1?  7 


of  the  piece,  and  ap  the  alti- 
tude due  to  the  projectile  velo- 
ci  H-  at  a 

fly  the  last  proposition,  find 
the  horizontal  range  ah  to  the 
given  velocity  and  direction  ; 
draw  he  perpendicular  to  ah, 
meeting  the  oblique  plane  in  e ; 
draw  ef  parallel  to  ag,  and 
fi  parallel  to  he  ; so  shall  the 
projectile  pass  through  i,  and  the  range  on  the  oblique 
plane  will  be  ai.  As  is  evident  by  theor  15  of  the  Para- 
bola, where  it  is  proved,  that  if  ah,  ai  be  any  two  lines  ter- 
minated at  the  curve,  and  if,  he  parallel  to  the  axis  ; then 
is  ef  parallel  to  the  tangent  ag. 

94.  Otherwise,  without  the  Horizontal  Range. 

Draw  pq  perp.  to  ag,  and  cid  perp.  to  the  horizontal 
plane  af,  meeting  the  inclined  plane  in  k ; take  ae  = 4ak, 
draw  ef  parallel  to  ag,  and  fi  parallel  to  ap  or  dq,  ; so  shall 
be  the  range  on  the  oblique  plane  For  ah  = 4ad, 
therefore  eh  is  parallel  to  fi,  and  so  on,  as  above. 

Otherwise. 

95.  Draw  p q making  the  angle  ap q = the  angle  gai  ; 
'bon  take  ag  = 4a<js  and  draw  gi  perp.  to  ah.  Or,  draw 

perp.  to  ah,  and  take  ai  = 4a k.  Also  k q will  be  equdl 
cv  thq  greatest  height  above  the  plane. 

For,  by  cor.  2,  prop.  20,  ap  : ag  : : ag  : 4gi 
and  by  sim.  triangles,  ap  : ag  : : Aq  : gi, 
or  - - - ap  : ag  : : 4Aq  : 4gi  ; 

therefore  ag  = 4a^  ; and  by  sim  triangles,  at  = 4a k. 

Also  qk,  or  Agi,  is  = to  cv  by  theor.  13  of  the  Parabola. 


138 


©F  MOTION,  FORCES,  &c. 


ap  be  bisected  by  the  perpendicular  sto  ; then  with  the 
centre  o describing  a circle  through  a and  p,  the  same  will 
also  pass  through  q.  because  the  angle  gai,  formed  by  the 
tangent  ai  and  ag,  is  equal  to  the  angle  af q,  which  will 
therefore  stand  on  the  same  arc  a q. 

97.  Carol.  2.  If  there  be  given  the  range  ai  and  the  ve- 
locity, or  the  impetus,  the  direction  will  hence  be  easily 
found  thus  : Take  a k = Aai,  draw  kq  perp.  to  ah,  meeting 
the  circle  described  with  the  radius  ao  in  two  points  q and 
q ; then  a q or  a q will  be  the  direction  of  the  piece  And 
hence  it  appears  that  there  are  two  directions,  which,  with 
the  same  impetus,  give  the  very  same  range  ai.  And  these 
two  directions  make  equal  angles  with  ai  and  ap,  because 
the  arc  p q is  equal  the  arc  a q.  They  also  make  equal  angles 
with  a line  drawn  from  a through  s,  because  the  arc  s q is 
equal  the  arc  s q. 

98.  Corot.  3.  Or,  if  there  be  given  the  range  ai,  and  the 

direction  a q ; to  find  the  velocity  or  impetus.  TakeAfc  = 
i ai,  and  erect  kq  perp.  to  ah,  meeting  the  line  of  direction 
in  q ; then  draw  q p making  the  Z.  Aqp  ~ ; so- shall 

ap  be  the  impetus,  or  the  altitude  due  to  the  projectile 
velocity. 

99.  Corol.  4.  The  range  on  an  oblique  plane,  with  a given 
elevation  is  directly  proportional  to  the  rectangle  of  the 
cosine  of  the  direction  of  the  piece  above  the  horizon,  and 
the  sine  of  the  direction  above  the  oblique  plane,  and  reci- 
procally to  the  square  of  the  cosine  of  the  angle  of  the  plane 
above  or  below  the  horizon 

For,  put  s = sin.  Z?AI  or  ap q, 

c = cos.  Z?AH  or  sin.  PA^» 
c — cos  ^'ah  or  sin.  a kd  or  a kq  or  &q?- 

Then,  in  the  triangle  ap q.  c s : : ap  : Aq  ; 
and  in  the  triangle  Akq,  c : c : : Aq  : Ak  ; 
theref  by  composition,  c2  : cs  : : ap  : ak  = {ai 

So  that  the  oblique  range  ai  = -^  X 4ap. 

100.  The  range  is  the  greatest  when  Ak  is  the  greatest  ; 
that  is  fwhen  kq  touches  the  circle  in  the  middle  point  s ; 
and  then  the  line  of  direction  passes  through  s,  and  bisects 
the,  angle  formed  by  the  oblique  plane  and  the  vertex.  Also, 
the  ranges  are  equal  at  equal  angles  above  and  below  this 
direction  for  the  maximum. 

101.  Corol.  5.  The  greatest  height  cv  or  kq  of  the  projec- 

tile, 


PROJECTILES. 


139 


$2 

®le,  above  the  plane,  is  equal  to  — X ap.  And  therefore  it 

is  as  the  impetus  and  square  of  the  sine  of  direction  above 
the  plane  directly,  and  square  of  the  cosine  of  the  plane’s  in* 
clination  reciprocally. 

For  - c (sin.  Aq p)  : s (sin.  ap q~)  : : ap  : a q, 
and  c (sin.  xkq ) : s (sin.  kAq ) : : a q : kq, 
theref.  by  comp  c2  : s2  : : ap  : kq 

102.  Corol.  6.  The  time  of  flight  in  the  curve  at>i  is  =: 

— */—  where  g = 16TV  feet.  And  therefore  it  is  as  the 

velocity  and  sine  of  direction  above  the  plane  directly,  and 
cosine  of  the  plane’s  inclination  reciprocally.  For  the  time 
of  describing  the  curve,  is  equal  to  the  time  of  falling  freely 
2 

through  gi  or  4 kq  or X ap.  Therefore,  the  time  being 

c2 

as  the  square  root  of  the  distance, 

g : ~\/  ap  : ; l'  the  t*me  flight. 

SCHOLIUM. 

103.  From  the  foregoing  corollaries  may  be  collected  the 
following  set  of  theorems  relating  to  projects  made  on  any 
given  inclined  planes,  either  above  or  below  the  horizontal 
plane.  In  which  the  letters  denote  as  before,  namely, 

c = cos.  of  direction  above  the  horizon, 
c = cos  of  inclination  of  the  plane, 
s = sin.  of  direction  above  the  plane, 

R the  range  on  the  oblique  plane, 

t the  time  of  flight, 

v the  projectile  velocity, 

h the  greatest  height  above  the  plane, 

a the  impetus,  or  alt.  due  to  the  velocity  v, 

g=  16^  feet.  Then, 


C6-  C9  src  „ 

= — X4a=  —v2  — - — t2  = 


4c 


C2 
S2 

h = — a 
c2 

v = ^/  4 ag 


c2,g 

S2  V 2 

4gC2 

c/' 

v Cs 


s 

JR 

4 C 
_gc 


H. 


— — T2  . 


2s  a «v 

c ^ g ec 


s 

g_ 

4 
2c 

= -v/  SH- 


*11  n H 

=V— = 

ffC 

And  frovi  any  of  these,  the  angle  of  direction  may  be  found. 

PRAG- 


14ft 


OP  MOTION,  FORCES,  &c. 


PRACTICAL  GUNNERY. 

104.  THE  two  foregoing  propositions  contain  the  whole 
theory  of  projectiles,  with  theorems  for  all  the  cases,  regu- 
larly arranged  for  use,  both  for  oblique  and  horizontal  planes. 
But  before  they  can  be  applied  to  use  in  resolving  the  several 
cases  in  the  practice  of  gunnery,  it  is  necessary  that  some 
more  data  be  laid  down,  as  derived  from  good  experiments 
made  with  balls  or  shells  discharged  from  cannon  or  mortars, 
by  gunpowder,  under  different  circumstances.  For,  without 
such  experiments  and  data,  those  theorems  can  be  of  very 
little  use  in  real  practice,  on  account  of  the  imperfections  and 
irregularities  in  the  tiring  of  gunpowder,  and  the  expulsion 
of  balls  from  guns,  but  more  especially  on  account  of  the 
enormous  resistance  of  the  air  to  all  projectiles  made  with 
any  velocities  that  are  considerable.  As  to  the  cases  in 
which  projectiles  are  made  with  small  velocities,  or  such  as 
do  not  exceed  200,  or  300,  or  400  feet  per  second  of  time, 
they  may  be  resolved  tolerably  near  the  truth,  especially  for 
the  larger  shells,  by  the  parabolic  theory,  laid  down  above. 
But,  in  cases  of  great  projectile  velocities,  that  theory  is  quite 
inadequate,  without  the  aid  of  several  data  drawn  from  many 
and  good  experiments.  For  so  great  is  the  effect  of  the  re- 
sistance of  the  air  to  projectiles  of  considerable  velocity,  that 
some  of  those  which  in  the  air  range  only  between  2 and  3 
miles  at  the  most,  would  in  vacuo  range  about  ten  times  as 
far,  or  between  20  and  30  miles. 

The  effects  of  this  resistance  are  also  various,  according  to 
the  velocity,  the  diameter,  and  the  weight  of  the  projectile. 
So  that  the  experiments  made  with  one  size  of  ball  or  shell, 
will  not  serve  for  another  size,  though  the  velocity  should  be 
the  same  ; neither  will  the  experiments  made  with  one  ve- 
locity, serve  for  other  velocities,  though  the  ball  be  the  same. 
And  therefore  it  is  plain  that,  to  form  proper  rules  for  prac- 
tical gunnery,  we  ought  to  have  good  experiments  made  with 
each  size  of  mortar,  and  with  every  variety  of  charge,  from 
the  least  to  the  greatest.  And  not  only  so,  but  these  ought 
also  to  be  repeated  at  many  different  angles  of  elevation 
namely  for  every  single  degree  between  30°  and  60°  elevation, 
and  at  intervals  of  5°  above  60°  and  below7  30,  from  the  ver- 
tical direction  to  point  blank.  By  such  a course  of  exper- 
iments it  will  be  found,  that  the  greatest  range,  instead  of  be- 
ing constantly  that  at  an  elevation  of  45°,  as  in  the  parabolic 
theory,  will  be  at  all  intermediate  degrees  between  45  and  30  ; 
'•••  !\  being 


PRACTICAL  GUNNERY. 


141 


being  more  or  less,  both  according  to  the  velocity  and  the 
weight  of  the  projectile  ; the  smaller  velocities  and  larger 
shells  ranging  farthest  when  projected  almost  at  an  elevation 
of  45°  ; while  the  greatest  velocities,  especially  with  the 
smaller  shells,  range  farthest  with  an  elevation  of  about  30°. 

105.  There  have,  at  different  times,  been  made  certain 
small  parts  of  such  a course  of  experiments  as  is  hinted  at 
above.  Such  as  the  experiments  or  practice  carried  on  in 
the  year  1773,  on  Woolwich  Common  ; in  which  all  the  sizes 
of  mortars  were  used,  and  a variety  of  small  charges  of  pow- 
der. But  they  were  all  at  the  elevation  of  45°  ; consequent- 
ly these  are  defective  in  the  higher  charges,  and  in  all  the 
ether  angles  of  elevation. 

Other  experiments  were  also  carried  on  in  the  same  place 
in  the.  years  1784  and  1786,  with  various  angles  of  elevation 
indeed,  but  with  only  one  size  of  mortar,  and  only  one  charge 
of  powder,  and  that  but  a small  one  too  ; so  that  all  those 
nearly  agree  with  the  parabolic  theory.  Other  experiments 
have  also  been  carried  on  with  the  ballistic  pendulum,  at  dif- 
ferent times  ; from  which  have  been  obtained  some  of  the  laws 
for  the  quantity  of  powder,  the  weight  and  velocity  of  the 
ball,  the  length  of  the  gun,  &c.  Namely,  that  the  velocity  of 
the  ball  varies  as  the  square  root  of  the  charge  directly,  and 
as  the  square  root  of  the  weight  of  ball  reciprocally  ; and 
that,  some  rounds  being  fired  with  a medium  length  of  one- 
pounder  gun,  at  1 5°  and  45°  elevation,  and  with  2,4,  8, 
and  12  ounces  of  powder,  gave  nearly  the  velocities,  ranges, 
and  times  of  flight,  as  they  are  here  set  down  in  the  fallowing 
Table. 


Powder. 

Elevation 
of  gun. 

Velocity 
of  ball. 

Range. 

Time  of 
flight. 

oz. 

feet. 

feet. 

2 

15° 

860 

4100 

9'' 

4 

15 

1 230 

5100 

12 

§ 

15 

1640 

6000 

14i 

12 

15 

1680 

6700 

15| 

2 

45 

860 

5100 

21 

106.  But  as  we  are  not  yet  provided  with  a sufficient 
number  and  variety  of  experiments,  on  .which  to  establish 
true  rules  for  practical  gunnery  independent  of  the  parabolic 
theory,  we  must  at  present  content  ourselves  with  the  data  of 

some 


142 


OF  MOTION,  FORCES,  &c. 


some  one  certain  experimented  range  and  time  of  flight,  at  & 
given  angle  of  elevation  ; and  then  by  help  of  these,  and  the 
rules  in  the  parabolic  theory,  determine  the  like  circumstances 
for  other  elevations  that  are  not  greatly  different  from  the 
former,  assisted  by  the  following  practical  rules. — 

SOME  PRACTICAL  RULES  IN  GUNNERY. 

I.  To  find  the  Velocity  of  any  Shot  or  Shell. 

Rule.  Divide  double  the  weight  of  the  charge  of  powder 
by  the  weight  of  the  shot,  both  in  lbs.  Extract  the  square  root 
of  the  quotient.  Multiply  that  root  by  1600,  and  the  product 
will  be  the  velocity  in  feet,  or  the  number  of  feet  the  shot 
passes  over  per  second. 

Or  say — As  the  root  of  the  weight  of  the  shot,  is  to  the  root 
of  double  the  weight  of  the  powder,  so  is  1600  feet,  to  the 
velocity. 

II.  Given  the  range  at  one  Elevation ; to  find  the  Range  at 

Another  Elevation. 

Rule.  As  the  sine  of  double  the  first  elevation,  is  to  its 
range  ; so  is  the  sine  of  double  another  elevation,  to  its 
range. 

III.  Given  the  Range  for  One  Charge  ; to  find  the  Range  for 

Another  Charge,  or  the  Charge  for  Another  Range. 

Rule.  The  ranges  have  the  same  proportion  as  the  charg- 
es ; that  is,  as  one  range  is  to  its  charge,  so  is  any  other  range 
to  its  charge  : the  elevation  of  the  piece  being  the  same  in 
both  cases. 

107.  Example  1.  If  a ball  of  1 lb.  acquire  a velocity  of 
1600  feet  per  second,  when  fired  with  8 ounces  of  powder  : 
it  is  required  to  find  with  what  velocity  each  of  the  several 
kinds  of  shells  will  be  discharged  by  the  lull  charges  of  pow- 
der, viz. 


Nature  of  the  shells  in  inches 

13 

10 

8 

Their  weight  in  lbs. 

196 

90 

48 

16 

8 

Charge  of  powder  in  lbs. 

9 

4 

2 

1 

1 

Ans.  The  velocities  are 

485 

477 

462 

566 

566 

108.  Exam.  2-  If  a shell  be  found  to  range  1000  yards, 
when  discharged  at  an  elevation  of  45°  ; how  far  will  it 

range 


PRACTICAL  GUNNERY. 


143 


range  when  the  elevation  is  30»  16',  the  charge  of  powder 
being  the  same  ? Ans.  2612  feet,  or  871  yards. 

109.  Exam.  3 The  range  of  a shell,  at  45°  elevation, 
being  found  to  be  3750  feet ; at  what  elevation  must  the  piece 
be  set,  to  strike  an  object  at  the  distance  of  2810  feet,  with 
the  same  charge  of  powder  ? 

Ans.  at  24°  16'  or  at  65°  44'. 

110  Exam.  4.  With  what  impetus,  velocity,  and  charge 
of  powder,  must  a 13-inch  shell  be  fired,  at  an  elevation  of 
32°  12',  to  strike  an  object  at  the  distance  of  3250  feet  ? 

Ans.  impetus  1802,  veloc.  340,  charge  4lb.  7ioz. 

111.  Exam.  5.  A shell  being  found  to  range  3500  feet 

when  discharged  at  an  elevation  of  25°  12'  ; how  far  then 
will  it  range  at  an  elevation  of  36°  15'  with  the  same  charge 
of  powder  ? Ans.  4332  feet. 

112.  Exam.  6.  If,  with  a charge  of  91b.  of  powder,  a shell 
range  4000  feet ; what  charge  will  suffice  to  throw  it  3000 
feet,  the  elevation  being  45°  in  both  cases  ? 

Ans.  6flb.  of  powder. 

113.  Exam.  7.  What  will  be  the  time  of  flight  for  any 
giveu  range,  at  the  elevation  of  45°  ? 

Ans.  the  time  in  secs,  is  £ the  sq  root  of  the  range  in  feet. 

114.  Exam.  8.  In  what  time  will  a shell  range  3250  feet, 

at  an  elevation  of  32°  ? Ans.  1 l^sec.  nearly. 

115.  Exam.  9.  How  far  will  a shot  range  on  a plane  which 
ascends  8°  15'  ; and  another  which  descends  8°  15'  ; the  im- 
petus being  3000  feet,  and  the  elevation  of  the  piece  32°  30'  ? 

Ans.  4244  feet  on  the  ascent, 
and  6745  feet  on  the  descent. 

116.  Exam.  10.  How  much  powder  will  throw  a 13-inch 
shell  4244-feet  on  an  inclined  plane,  which  ascends  8°  15', 
the  elevation  of  the  mortar  being  32^30'  ? 

Ans.  7,37651b,  or  7lb.  6oz. 

117.  Exam.  11.  At  what  elevation  must  a 13-inch  mortar 

be  pointed  to  range  6745  feet  on  a plane  which  descends 
8°  15' ; the  charge  7|lb,  of  powder  ? Ans.  32°  28'. 

118.  Exam.  12.  In  what  time  will  a 13-inch  shell  strike  a 

plane  which  rises  8°  30',  when  elevated  45°,  and  discharged 
with  an  impetus  of  2304  feet  ’ Ans.  14f  seconds. 

THE 


L 144  J 

THE  DESCENT  OF  BODIES  ON  INCLINED  PLANES 
AND  CURVE  SURFACES. — T HE  MOTION  OF  PEN- 
DULUMS. 


PROPOSITION  xxm. 


119.  If  a weight  w he  Sustained  on  an  Inclined  Plane  ab  by  a 
Power  p,  acting  in  a Direction  wp,  Parallel  to  the  Plane,  Then 


The  Length  ab. 

The  Height  bc,  and 
The  Base  ac, 
of  the  Plane. 


The  Weight  of  the  Body , w 
The  Sustaining  Power  p,  and 
The  Pressure  on  the  Plane,  p, 
are  respectively  as 
For,  draw  cd  perpendicular 
to  the  plane.  Now  here  are 
three  forces,  keeping  one  an- 
other in  equilibrio  ; namely,  the 
weight,  or  force  of  gravity,  act- 
ing perpendicular  to  ac,  or  pa- 
rallel to  bc  ; the  power  acting 
parallel  to  db  ; and  the  pressure  perpendicular  to  ab,  or  pa- 
rallel to  dc  : but  when  three  forces  keep  one  another  in  equili- 
brio, they  are  proportional  to  the  sides  of  the  triangle  cbb, 
made  by  lines  in  the  direction  of  those  forces,  by  prop.  8 ; 
therefore  those  forces  are  to  one  another  as  bc,  bd,  cd.  But 
the  two  triangles  abc,  cbd,  are  equiangular,  and  have  their 
like  sides  proportional  ; therefore  the  three  bd,  bc,  cd,  are  to 
one  another  respectively  as  the  three  ab,  bc,  ac  ; which  there- 
fore are  as  the  three  forces  w,  p ,p. 

120.  Corol.  1.  Hence  the  weight  w,  power  p,  and  pressure 
p,  are  respectively  as  radius,  sine  and  cosine, 

of  the  plane’s  elevation  bac  above  the  horizon. 


For,  since  the  sides  of  triangles  are  as  the  sines  of  their 
opposite  angles,  therefore  the  three  ab,  bc,  ac, 
are  respectively  as  - - - sin  c,  sin.  a,  sin  b, 

or  as  - - - - - radius,  sine,  cosine, 

of  the  angle  a of  elevation. 

Or,  the  three  forces  are  as  ac,  cd,  ad  ; perpendicular  to 
their  directions. 

121.  Corol.  2.  The  power  or  relative  weight  that  urges  a 
body  w down  the  inclined  plane  is  = ~ X w ; or  the  force 

with 


DESCENTS  ON  INCLINED  PLANES. 


145 


with  which  it  descends,  or  endeavours  to  descend,  is  as  the 
sine  of  the  angle  a of  inclination. 


122.  Corol.  3.  Hence,  if  there  be 
two  planes  of  the  same  height,  and 
two  bodies  be  laid  on  them  which 
are  proportional  to  the  lengths  of 
the  plane-  they  will  have  an  equal 
tendency  to  descend  down  the  planes. 

And  consequently  they  will  mutually  sustain  each  other  if 
they  be  connected  by  a string  acting  parallel  to  the  planes. 


123.  Corol.  4.  In  like  manner, 
when  the  power  p acts  in  any 
other  direction  whatever,  wp  ; by 
drawing  cde  perpendicular  to  the 
direction  wp,  the  three  forces  in 
equilibrio,  namely,  the  weight  w, 
the  power  p,  and  the  pressure  on 
the  plane,  will  still  he  respectively 
as  ac,  cd,  ad,  drawn  perpendicular 
t«  the  direction  of  those  forces. 


PROPOSITION  XXIV. 


124.  If  a Weight  w on  an  Inclined  Plane  ab,  he  in  Equilibrio 
with  another  Weight  p hanging  freely ; then  if  they  be  set 
a-moving,  their  Perpendicular  Velocities,  in  that  Place,  will 
he  Reciprocally  as  those  Weights. 

Let  the  weight  w descend  a very  I? 

small  space,  from  w to  a,  along  the 
plane,  by  which  the  string  pfw  will 
come  into  the  position  ffa.  Draw 
wh  perpendicular  to  the  horizon  ac, 
and  wg  perpendicular  to  af  : then 
wh  will  be  the  space  perpendicularly 
descended  by  the  weight  w ; and  a», 
or  the  difference  between  fa  and  fw, 
will  be  the  space  perpendicularly  ascended  by  the  weight  p • 
and  their  perpendicular  velocities  are  as  those  spaces  wh 
and  ag  passed  over  in  those  directions,  in  the  same  time. 
Draw  cde  perpendicular  to  af,  and  di  perpendicular  to  ac. 
Then, 

in  the  sim.  figs,  agwh  and  aedi,  ag  : wh  : : ae  : di  ; 

and  in  the  sim.  tri.  aec,  dic,  ac  : cd  : : ae  : di  j 

but,  by  cor.  4,  prop.  23,  ac  : cd  : : w : p ; 

therefore,  by  equality,  ag  : wh  : : w : p ; 

Vol.  If.  ' 20  That 


20 


46 


OF  MOTION,  FORCES,  &c. 


That  is,  their  perpendicular  spaces,  or  velocities,  are  re- 
ciprocally  as  their  weights  or  masses. 

126.  Corol.  1.  Hence  it  follows,  that  if  any  two  bodies  be 
in  equilibrio  on  two  inclined  planes,  and  if  they  be  set  a- 
moving,  their  perpendicular  velocity  will  be  reciprocally  as 
their  weights.  Because  the  perpendicular  weight  which  sus- 
tains the  one,  -would  also  sustain  the  other. 

126  Corol.  2 And  hence  also,  if  two  bodies  sustain  each 
other  in  equilibrio,  on  any  planes,  and  they  be  put  in  motion  ; 
then  each  body  multiplied  by  its  perpendicular  velocity,  will 
give  equal  products. 

PROPOSITION  XXV. 

127.  The  Velocity  acquired  by  a Body  descending  freely  down 

an  Inclined  Plane  ab,  is  to  the  Velocity  acquired  by  a Body 
falling  Perpendicularly,  in  the  same  Time  ; as  the  Height  of 

the  Plane  bc,  is  to  its  Length  ab. 

For  the  force  of  gravity,  both  per- 
pendicularly and  on  the  plane,  is  con- 
stant ; and  these  two,  by  corol.  2,  prop. 

23,  are  to  each  other  as  ab  to  bc.  But, 
by  art.  28,  the  velocities  generated  by 
any  constant  forces,  in  the  same  time, 
are  as  those  forces.  Therefore  the  velocity  down  ba  is  t* 
the  velocity  down  bc,  in  the.  same  time,  as  the  force  on  ba  to 
the  force  on  bc  : that  is,  as  bc  to  ba. 

128.  Corol.  1.  Hence,  as  the  motion  down  an  inclined 
plane  is  produced  by  a constant  force,  it  will  be  a motion 
uniformly  accelerated  ; and  therefore  the  laws  before  laid 
down  for  accelerated  motions  in  general,  hold  good  for  nio- 
tions  on  inclined  planes  ; such,  for  instance,  as  the  following  : 
That  the  velocities  are  as  the  times  of  descending  from  rest ; 
that  the  spaces  descended  are  as  the  squares  of  the  velocities, 
or  squares  of  the  times  ; and  that  if  a body'  be  thrown  up  an 
inclined  plane,  with  the  velocity  it  acquired  in  descending,  it 
will  lose  all  its  motion,  and  ascend  to  the  same  height,  in  the 
same  time,  and  will  repass  any  point  of  the  plane  with  the 
same  velocity  as  it  passed  it  in  descending. 

129.  Corol.  2.  Hence  also,  the  space  descet»ded  down  an 
inclined  plane,  is  to  the  space  descended  perpendicularly,  in 
the  same  time,  as  the  height  of  the  plane  cfe,  to  its  length 
ab,  or  as  the  sine  of  inclination  to  radius.  For  the  spaces 

described 


DESCENTS  ON  INCLINED  PLANES. 


147 


described  by  any  forces,  in  the  same  time,  are  as  the  forces,  or 
as  the  velocities. 

130.  Corol.  3.  Consequently  the  velocities  and  spaces  de- 
scended by  bodies  down  different  inclined  planes,  are  as  the 
sines  of  elevation  of  the  planes. 

131.  Corol.  4.  If  cd  be  drawn  perpendicular  to  ab  ; then 
while  a body  falls  freely  through  the  perpendicular  space  bc, 
another  body  will  in  the  same  time,  descend  down  the  part  of 
of  the  plaue  bd.  For  by  similar  triangles,  - 

bc  : bd  : : ba  : bc,  that  is,  as  the  space  descended,  by  co- 
rol. 2. 

Or,  in  any  right-angled  triangle  bdc, 
having  its  hypothenuse  bc  perpendicular 
to  the  horizon,  a body  will  descend  down 
any  of  its  three  sides  bd,  bc,  dc,  in  the 
same  time.  And  therefore,  if  on  the  dia- 
meter bc  a circle  be  described,  the  time 
of  descending  down  any  chords  bd.be,  bf, 
dc,  ec,  fc,  &c.  will  be  all  equal,  and  each 
equal  to  the  time  of  falling  freely  tnrough 
the  perpendicular  diameter  bc. 

PROPOSITION  XXVI. 

132.  The  Time  of  descending  down  the  inclined  Plane  ba,  is  to 

the  Time  of  falling  through  the  Height  of  the  Plane  bc,  as  the 

Length  ba  is  to  the  Height  bc. 

Draw  cd  perpendicular  to  ab. 

Then  the  times  of  describing  bd  and 
bc  are  equal  by  the  last  corol.  Call 
that  time  t,  and  the  time  of  describ- 
ing ba  call  T. 

Now,  because  the  space  describ- 
ed by  constant  forces,  are  as  the  squares  of  the  times  ; there- 
fore t2  : t2  : : bd  : ba. 

But  the  three  bo,  bc,  ba,  are  in  continual  proportion  : 
therefore,  bd  : ba  : : bo2  : : ba2  ; 
hence,  by  equality,  t2  : t2  : : bc2  : ba2, 
or  - - t : t : : bc  : ba. 

133.  Corol.  Hence  the  times  .of  descending  down  different 
planes  of  the  same  height,  are  to  one  another  aa  the  lengths 
of  the  planes. 


PROPOSITION 


148 


OP  MOTION,  FORCES,  &c. 


PROPOSITION  XXVII. 

334.  A Body  acquires  the  Same  Velocity  in  descending  down  any 
Inclined  Plane  ba,  as  by  falling  perpendicular  through  the 
Height  of  the  Plane  bc. 

For,  the  velocities  generated  by  any  constant  forces,  are  in 
the  compound  ratio  of  the  forces  and  times  of  acting 
But  if  we  put 

f to  denote  the  whole  force  of  gravity  in  bc, 

/ the  force  on  the  plane  ab, 
t the  time  of  describing  bc,  and 
t the  time  of  descending  down  ab  ; 

. then  by  art.  1 19,  f : f : : ba  : bc  ; 
and  by  art.  132,  t,  : t : : bc  : ba  ; 
theref.  by  comp.  Ft  :fr  : : 1 : 1. 

That  is  the  compound  ratio  of  the  forces  and  times,  or  the 
ratio  of  the  velocities,  is  a ratio  of  equality. 

135.  Corol  1.  Hence  the  velocities  acquired,  by  bodies 
descending  down  any  planes,  from  the  same  height,  to  the 
same  horizontal  line  are  equal. 

136.  Corol.  2.  If  the  velocities  be  equal,  at  any  two  equal 
altitudes,  d,  e ; they  will  be  equal  at  all  other  equal  altitudes 
a,  c. 

137.  Corol.  3.  Hence  also  the  velocities  acquired  by  de- 
scending down  any  planes,  are  as  the  square  roots  of  the 
heights. 

PROPOSITION  XXVIII. 

138.  If  a Body  descend  down  any  Number  of  Contiguous  Planes. 
ab,  bc,  cd  ; it  will  at  last  acquire  the  Same  Velocity,  as  a Body 
falling  perpendicularly  through  the  Same  Height  ed,  supposing 
the  Velocity  not  altered  by  changing  from  one  Plane  to  an- 
other. 

Produce  the  planes  dc,  cb,  to 
meet  the  horizontal  line  ea  pro- 
duced in  f and  g.  Then,  by 
art.  135,  the  velocity  at  b is  the 
same  whether  the  body  descend 
through  ab  or  fb.  And  therefore 
the  velocity  at  c will  be  the  same, 
whether  the  body  descend  through  abc  or  through  fc 

which 


E AFC 


DESCENTS  ON  INCLINED  PLANES. 


149 


which  is  also  again,  by  art.  135,  the  same  as  by  descending 
through  gc.  Consequently  it  will  have  the  same  velocity 
at  d,  by  descending  through  the  planes  ab,  bc,  cd,  as  by  de- 
scending through  the  plane  gd  ; supposing  no  obstruction 
to  the  motion  by  the  body  impinging  on  the  planes  at  b and 
c : and  this  again,  is  the  same  velocity  as  by  descending 
through  the  same  perpendicular  height  ed. 

139.  Corol.  1.  If  the  lines  abcd,  &c.  be  supposed  inde- 
finitely small,  they  will  form  a curve  line,  which  will  be  the 
path  of  the  body  ; from  which  it  appears  that  a body  ac- 
quires also  the  same  velocity  in  descending  along  any  curve, 
as  in  falliag  perpendicularly  through  the  same  height. 

140.  Corol.  2.  Hence  also,  bodies  acquire  the  same  velo- 
city by  descending  from  the  same  height,  whether  they 
descend  perpendicularly,  or  down  any  planes,  or  down  any 
curve  or  curves.  And  if  their  velocities  be  equal,  at  any  one 
height,  they  will  be  equal  at  all  other  equal  heights.  There- 
fore the  velocity  acquired  by  descending  down  any  lines  or 
curves,  are  as  the  square  roots  of  the  perpendicular  heights. 

141.  Corol.  3.  And  a body,  after  its  descent  through  any 
curve,  will  acquire  a velocity  which  will  carry  it  to  the  same 
height  through  an  equal  curve,  or  through  any  other  curve 
either  by  running  up  the  smooth  concave  side,  or  by  being 
retained  in  the  curve  by  a string,  and  vibrating  likq  a pen- 
dulum : Also,  the  velocities  will  be  equal,  at  all  equal  alti- 
tudes ; and  the  ascent  and  descent  will  be  performed  in  the 
same  time,  if  the  curyes  be  the  same. 

PROPOSITION  XXIX. 

142.  The  Times  in  which  Bodies  descend  through  Similar  Parts 

of  Similar  Curves,  abc,  abc,  placed  alike,  are  as  the  Square 

Roots  of  their  Lengths. 

That  is,  the  time  in  ac  is  to  the  time  in  ac,  as  ac 
to  ac. 

For,  as  the  curves  are  similar,  they  may 
be  considered  as  made  up  of  an  equal 
number  of  corresponding  parts,  which  are 
every  where,  each  to  each,  proportional  to 
the  whole.  And  as  they  are  placed  alike, 
the  corresponding  small  similar  parts  will 
also  be  parallel  to  each  other.  But  the 
time  of  describing  each  of  these  pairs  of  corresponding  pa- 
rallel parts,  by  art.  128,  are  as  the  square  roots  of  their 

lengths. 


150 


©F  MOTION,  FORCES,  kc. 


lengths,  which  by  the  supposition,  are  as  y/  ac  to  y/  ac,  the 
roots  of  the  whole  curves  Therefore,  the  whole  times  are 
in  the  same  ratio  of  ac  to  y/  ac. 

143.  Corol.  1.  Because  the  axes  dc,  dc,  of  similar  curves, 
are  as  the  lengths  of  the  similar  parts  ac,  ac  ; therefore  the 
times  of  descent  in  the  curves  ac,  ac,  are  as  y/  dc  to  y/  dc, 
or  the  square  roots  of  their  axes. 

144.  Corol.  2.  As  it  is  the  same  thing,  whether  the  bodies 
run  down  the  smooth  concave  side  of  the  curves,  or  he  made 
to  describe  those  curves  by  vibrating  like  a pendulum,  the 
lengths  being  dc,  dc  ; therefore  the  times  of  the  vibration 
of  pendulums,  in  similar  arcs  of  any  curves,  are  as  the  square 
roots  of  the  lengths  of  the  pendulums. 

SCHOLIUM. 

145.  Having  in  the  last  corollary,  mentioned  the  pen- 
dulum, it  may  not  be  improper  here  to  add  some  remarks 
concerning  it. 

A pendulum  consists  of  a ball,  or  any 
other  heavy  body  b,  hung  by  a fine  string 
or  thread,  moveable  about  a centre  a, 
and  describing  the  arc  cbd  ; by  which 
vibration  the  same  motions  happen  to  this 
heavy  body,  as  would  happen  to  any 
body  descending  by  its  gravity  along  the 
spherical  superficies  cbd,  if  that  superfi- 
cies were  perfectly  hard  and  smooth.  If  the  pendulum  be 
carried  to  the  situation  ac,  and  then  let  fall,  the  ball  in  de- 
scending will  describe  the  arc  cb  ; and  in  the  point  b it 
will  have  that  velocity  which  is  acquired  by  descending 
through  cb,  or  by  a body  falling  freety  through  eb.  This 
velocity  will  be  sufficient  to  cause  the  ball  to  ascend  through 
an  equal  arc  bd,  to  the  same  height  d from  whence  it  fell 
at  c ; having  there  lost  all  its  motion,  it  will  again  begin  to 
descend  by  its  own  gravity  ; and  in  the  lowest  point  b it  will 
acquire  the  same  velocity  as  before  ; which  will  cause  it  to 
re-ascend  to  c ; and  thus,  by  ascending  and  descending,  it  will 
perform  continual  vibrations  on  the  circumference  cbd.  And 
if  the  motions  of  pendulums  met  with  no  resistance  from 
the  air,  and  if  there  were  no  friction  at  the  centre  of  mo- 
tion a,  the  vibrations  of  pendulums  would  never  cease. 
But  from  these  obstructions,  though  small,  it  happens,  that 
the  velocity  of  the  ball  in  the  point  b is  a little  diminished 
in  every  vibration ; and  consequently  it  does  not  return 
precisely  to  the  same  points  c or  d,  but  the  arcs  described  con- 
tinually 


PENDULUMS. 


151 


tinually  become  shorter  and  shorter,  till  at  length  they  are 
insensible  ; unless  the  motion  be  assisted  by  a mechanical 
contrivance,  as  in  clocks,  called  a maintaining  power. 

DEFINITION. 

146.  If  the  cir- 
cumference of  a 
circle  be  rolled  on 
a right  line,  begin- 
ing  at  any  point 
a,  and  continued 
till  the  same  point 
a arrive  at  the  line 
again,  making  just  one  revolution,  and  thereby  measuring 
out  a straight  line  aba  equal  to  the  circumference  of  the  cir- 
cle, while  the  point  a in  the  circumference  traces  out  a curve 
line  acaga  ; then  this  curve  is  called  a cycloid  ; and  some 
of  its  properties  are  contained  in  the  following  lemma. 

LEMMA. 

147.  If  the  generating  or  revolving  circle  be  placed  in  the 
middle  of  the  cycloid,  its  diameter  coinciding  with  the  axis 
ab,  and  from  any  point  there  be  drawn  the  tangent  cf,  the 
ordinate  cde  perp.  to  the  axis,  and  the  chord  of  the  circle 
ad  : Then  the  chief  properties  are  these  : 

The  right  line  cd  = the  circular  arc  ad  ; 

The  cycloidal  arc  ac  = double  the  chord  ad  ; 

The  semi-cyoloid  aca  — double  the  diameter  ab,  and 
The  tangent  cf  is  parallel  to  the  chord  ad. 

PROPOSITION  XXX. 

148.  When  a Pendulum  vibrates  in  a cycloid  ; the  Time  of  one 
Vibration,  is  to  the  Time  in  which  a Body  falls  through  Half 
the  Length  of  the  Pendulum,  as  the  Circumference  of  a dr 
cle  is  to  its  Diameter. 

Let  ABa  be  the  cycloid  ; 
db  its  axis,  or  the  diameter 
of  the  generating  semicircle 
deb  ; cb  = 2db  the  length 
of  the  pendulum,  or  radius 
of  curvature  at  b.  Let  the 
ball  descend  from  f,  and, 
in  vibrating,  describe  the 
arc  fb f.  Divide  fb  into  in- 
numerable small  parts,  one 
of  which  is  g g ; draw  fel, 

«m,  gm,  perpendicular  to 


C 


B 


15« 


OF  MOTION,  FORCES,  &c. 


bb.  On  lb  describe  the  se- 
micircle lmb,  whose  cen- 
tre is  o ; draw  mp  parallel 
to  db  ; also  draw  the  chords 
be,  bh,  eh,  and  the  radius 

OM. 

Now  the  triangles  beh, 
bhk,  are  equiangular  ; there- 
fore be  : bh  : : bh  : be,  or 
BH2  = BK  . BE,  Or  BH  = y/ 

(bk  . be). 

And  the  equiangular  triangles  mmip,  mon,  give 

up  : Mm  : : jin  : mo.  Also,  by  the  nature  of  the  cycloid, 

h h is  equal  to  eg. 

If  another  body  descend  down  the  chord  eb,  it  will  have 
the  same  velocity  as  the  ball  in  the  cycloid  has  at  the  same 
height.  So  that  k k and  eg  are  passed  over  with  the  same 
velocity,  and  consequently  the  time  in  passing  them  will  be 
as  their  lengths  eg,  kA,  or  as  h h to  k A,  or  bh.  to  bk  by 
similar  triangles,  or  y/  (bk  . be)  to  bk,  or  y/  be  to  y/  bk,  or  as 
y/  bl  to  y/  bn  by  similar  triangles. 

That  is,  the  time  in  g g : time  in  kA  : : y/  bl  : y/  bn. 

Again,  the  time  of  describing  any  space  with  a uniform 
motion,  is  directly  as  the  space,  and  reciprocally  as  the  ve- 
locity ; also,  the  velocity  in  k or  k A,  is  to  the  velocity  at  b, 
as  y/  ek  to  y/  eb,  or  as  y/  ln  to  y/  lb  ; and  the  uniform  ve- 
locity for  eb  is  equal  to  half  that  at  the  point  b,  therefore  the 

..  . . Kk  EB  Nn  LB 

time  in  kk  : time  in  eb  : : — - — : - — - — : : : 

y/LN  Sy/LB  y/LN  i y/LB 

(by  sim.  tri.)  : : ntj  or  mp  : 2y/  (bl  . ln.) 

That  is,  the  time  in  kA  : time  in  eb  : : mp  : 2y/  (bl  . ln.) 

But  it  was,  time  in  g g : time  in  kA  : : y/  bl  : y/  bn  ; theref. 
by  comp,  time  in  eg  : time  in  eb  : : mp  : 2y/  (bn  . nl)  or  2nm. 
But,  by  sim.  tri.  urn  : 2om  or  bl  : : mp  : 2nm. 

Theref.  time  in  cg  : time  in  eb  : : urn  : bl. 

Consequently  the  sum  of  all  the  times  in  all  the  Gg’s,  is  to 
the  time  in  eb,  or  the  time  in  db,  which  is  the  same  thing, 
as  the  sum  of  all  the  Mm’s,  is  to  lb  ; 
that  is,  the  time  in  f g : time  in  db  : : l m : lb, 
and  the  time  in  fb  : time  in  db  : : lmb  : lb, 
or  the  time  in  fb/ : time  in  db  : : 2lmb  : lb. 

That  is,  the  time  of  one  whole  vibration, 

is  to  the  time  of  falling  through  half  cb, 
as  the  circumference  of  any  circle, 
is  to  its  diameter. 


149.  Cor$i. 


PENDULUMS. 


153 


149.  Carol.  1.  Hence  all  the  vibrations  of  a pendulum  in 
a cycloid,  whether  great  or  small,  are  performed  in  the  same 
time,  which  time  is  to  the  time  of  falling  through  the  axis, 
or  half  the  length  of  the  pendulum,  as  3-1416,  to  1,  the  ratio 
of  the  circumference  to  its  diameter  ; and  hence  that  time 
is  easily  found  thus.  Put  p = 3-1416,  and  / the  length  of 
the  pendulum,  also  g the  space  fallen  by  a heavy  body  in 
1''  of  time. 

then  y'  g : y/\l  : : l"  : the  time  of  falling  through  }f, 

theref.  1 : p p */—,  which  therefore  is  the  time  of 

j v 2g  1 * 2 g 

one  vibration  of  the  pendulum. 

150.  And  if  the  pendulum  vibrate  in  a small  arc  of  a circle  ; 
because  that  small  arc  nearly  coincides  with  the  small  cy- 
cloidal arc  at  the  vertex  b ; therefore  the  time  of  vibration  in 
the  small  arc  of  a circle,  is  nearly  equal  to  the  time  of  vibra- 
tion in  the  cycloidal  arc  ; consequently  the  time  of  vibration 

in  a small  circular,  arc  is 

of  the  circle. 

151  So  that,  if  one  of  these,  g or  l,  be  found  by  experi- 
ment, this  theorem  will  give  the  other.  Thus,  if  g,  or  the 
space  fallen  through  by  a heavy  body  in  1"  of  time,  be  found, 
then  this  theorem  will  give  the  length  of  the  second  pendu- 
lum. Or,  if  the  length  of  the  second  pendulum  be  ob- 
served by  experiment,  which  is  the  easier  way,  this  theorem 
will  give  g the  descent  of  gravity  in  I'*.  Now,  in  the  lati- 
tude of  London,  the  length  of  a pendulum  which  vibrates 
seconds,  has  been  found  to  be  39}  inches  ; and  this  being 

39} 

written  for  l in  theorem,  it  gives  p = 1"  : hence  is 

. 2§ 
found  g = \p-  l = ip2  X 39}  = 193-07  inches  = 16Tu  feet, 
for  the  descent  of  gravity  in  1"  ; which  it  has  also  been 
found  to  be,  very  nearly,  by  many  accurate  experiments. 


equal  to  p where  l is  the  radius 


SCHOLIUM. 

152.  Hence  is  found  the  length  of  a pendulum  that  shall 
make  any  number  of  vibrations  in  a given  time.  Or,  the 
number  of  vibrations  that  shall  be  made  by  a pendulum  of 
a given  length.  Thus,  suppose  it  were  required  to  find  the 
length  of  a half-seconds  pendulum,  or  a quarter-seconds 
pendulum  ; that  is,  a pendulum,  to  vibrate  twice  in  a second, 
or  4 times  in  a second.  Then  since  the  time  of  vibration 
is  as  the  square  root  of  the  length, 

Vor..  IT-.  <2\ 


therefore 


MECHANICAL  POWERS'. 


therefore  1 : } : : ^/39\  : 1 , 

39} 

or  - * 1 : } : : 39}  : = 9|  inches  nearly,  the  length 

4 

of  the  half-seconds  pendulum.  Again  1 : T’F  : : 39}  : 2}  in- 
ches, the  length  of  the  quarter-seconds  pendulum. 

Again,  if  it  were  required  to  find  how  many  vibrations  a 
pendulum  of  80  inches  long  will  make  in  a minute.  Here 

29J- 

v/80  : v/39}  : : 60"  or  1'  : 60y 1 = 7}  y 313  = - - 

80 

41  95987,  or  almost  42  vibrations  in  a minute 

153.  In  these  propositions,  the  thread  is  supposed  to  be 
very  fine,  or  of  no  sensible  weight,  and  the  ball  very  small, 
<tfr  all  the  matter  united  in  one  point  ; also,  the  length  of 
the  pendulum,  is  the  distance  from  the  point  of  suspension, 
or  centre  of  motion,  to  this  point,  or  centre  of  the  small 
ball.  But  if  the  hall  be  large,  or  the  string  very  thick,  or 
the  vibrating  body  be  of  any  other  figure  ; then  the  length 
of  the  pendulum  is  different,  and  is  measured,  from  the 
centre  of  motion,  not  to  the  centre  of  magnitude  of  the 
body,  but  to  such  a point,  as  that  if  all  the  matter  of  the 
pendulum  were  collected  into  it,  it  would  then  vibrate  in 
the  same  time  as  the  compound  pendulum  ; and  this  point  is 
called  the  Centre  of  Oscillation  ; a point  which  will  be  treated 
of  in  what  follows. 

THE  MECHANICAL  POWERS,  kc. 

154.  WEIGHT  and  Power,  when  opposed  to  each  other, 
■Ignify  the  body  to  be  moved,  and  the  body  that  moves  it  , 
or  the  patient  and  agent.  The  power  is  the  agent,  which 
moves,  or  endeavours  to  move,  the  patient  or  weight. 

155.  Equilibrium,  is  an  equality  of  action  or  force,  be* 
tween  two  or  more  powers  or  weights,  acting  against  each 
other,  by  which  they  destroy  each  other’s  effects,  and  remain 
at  rest. 

156.  Machine,  or  Engine,  is  any  Mechanical  instrument 
contrived  to  move  bodies.  And  it  is  composed  of  the  me- 
chanical powers. 

157.  Mechanical  Powers,  are  certain  simple  instruments, 
commonly  employed  for  raising  greater  weights,  or  overcom- 
ing greater  resistances,  than  could  be  effected  by  the  natural 
strength  without  them.  These  are  usually  accounted  six  in 

number, 


THE  LEVER. 


155 


number,  viz.  the  Lever,  the  Wheel  and  Axle,  the  Pulley,  the 
Inclined  Plane,  the  Wedge,  and  the  Screw. 

158  Mechanics,  is  the  science  of  forces,  and  the  effects 
they  produce,  when  applied  to  machines,  in  the  motion  of 
bodies. 

15b.  Statics,  i?  the  science  of  weights,  especially  when 
considered  in  a state  of  equilibrium. 

160  Centre  of  Motion,  is  the  fixed  point  about  which  a 
body  moves.  And  the  Axis  of  Motion,  is  the  fixed  line  about 
which  it  moves. 

161.  Centre  of  Gravity,  is  a certain  point,  on  which  a body 
being  freely  suspended,  it  will  rest  in  any  position. 

— 

OF  THE  LEVER. 

162.  A Lever  is  any  inflexible  rod,  bar,  or  beam,  which 
serves  to  raise  weights,  while  it  is  supported  at  a point  by  a 
fulcrum  or  prop,  which  is  the  centre  of  motion.  The  lever 
is  supposed  to  be  void  of  gravity  or  weight,  to  render  the 
demonstrations  Gasier  and  simpler.  There  are  three  kinds 
•f  levers. 

163.  A Lever  of  the  First 
kind  has  the  prop  c be- 
tween the  weight  w and 
the  power  p.  And  of  this 
kind  are  balances,  scales, 

Grows,  hand-spikes,  scissors, 
pinchers,  & c. 

164.  A Lever,  of  the  Se- 
cond kind  has  the  weight 
between  the  power  and  the 
prop  Such  as  oars,  rud- 
ders, cutting  knives  that  are 
fixed  at  one  end,  &c. 

165.  A Lever  of  the 
Third  kind  has  the  power 
between  the  weight  and 
the  prop.  Such  as  tongs, 
the  bones  and  muscles  of 
animals,  a man  rearing  a 
ladder,  &c. 


166.  A 


156 


MECHANICAL  POWERS 


166.  A Fourth  kind  is  some- 
times added,  called  the  Bended  q 

Lever.  As  a hammer  drawing  w x 

a nail  ^ Q 

J67.  In  all  these  instruments  the  power  may  he  repre- 
sented by  a weight,  which  is  its  most  natural  measure,  acting 
downward  : but  having  its  direction  changed,  when  necessary, 
by  means  of  a fixed  pulley. 


PROPOSITION  XXXI. 

168.  When  the  Weight  and  Power  keep  the  Lever  in  Equilibria 
they  are  to  each  other  Reciprocally  as  the  Distances  of  their 
Lines  of  Direction  from  the  Prop.  That  is,  p : w : : cd  : 
ce  : where  cd  and  ce  are  perpendicular  to  wo  and  ao,  the 
Di  rections  of  the  two  Weights,  or  the  Weight  and  Power 
w and  a. 

For,  draw  cf  parallel  to  ao,  and 
cb  parallel  to  wo  : Also  join  co, 
which  will  be  tjie  direction  of  the 
pressure  on  the  prop  c ; for  there 
cannot  be  an  equilibrium  unless  the 
directions  of  the  three  forces  all  meet 
in,  or  tend  to,  the  same  point,  as  o. 

Then,  because  these  three  forces 
keep  each  other  in  equilibrio,  they 
are  proportional  to  the  sides  of  the 
triangle  cbo  or  cfo,  drawn  in  the 
direction  of  those  forces  ; there- 
fore - - - - p : w 

But,  because  of  the  parallels,  the 
two  triangles  cdf,  ceb  are  equiangu- 
lar, therefore  - cd 

Hence,  by  equality,  - r 

That  is  each  force  is  reciprocally  proportional 
distance  of  its  direction  from  the  fulcrum. 

And  it  will  be  found  that  this  demonstration  will  serve  for 
all  the  other  kinds  of  levers,  by  drawing  the  lines  as  directed. 

169.  Corol.  1.  When  the  angle  a is  ==  the  angle  w,  then 
is  cd  : ce  : : cw  : ca  : : p : w.  Or  when  the  two  forces  act 
perpendicularly  on  the  lever,  as  two  weights,  &c.  ; then,  in 
case  of  an  equilibrium,  d coincides  with  w,  and  e with  p ; 
consequently  then  the  above  proportion  becomes  also  p : w : : 
cw  : ga,  or  the  distances  of  the  two  forces  from  the  fulcrum, 
taken  on  the  lever,  are  reciprocally  proportional  to  those 
forces 

170  Carol 


cf  : fo  or  cb- 


CF 

CD 


CB. 

CE. 

to  the 


THE  LEVER. 


157 


170.  Corol.  2.  If  any 'force  p be  applied  to  a lever  at  a ; its 
effect  on  the  lever,  to  turn  it  about  the  centre  of  motion  c,  is 
as  the  length  of  the  lever  ca.  and  the  sine  of  the  angle  of  di- 
rection cae.  For  the  perp.  ce  is  as  ca  X s /_  a. 

171.  Corol.  3.  Because  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means,  therefore  the  product  of 
the  power  by  the  distance  of  its  direction,  is  equal  to  the  pro- 
duct of  the  weight  by  the  distance  of  its  direction. 

That  is,  p X ce  = w X cd. 

172.  Corol.  4.  If  the  lever,  with  the  weight  and  power 
fixed  to  it,  be  made  to  move  about  the  centre  c ; the  momen- 
tum of  the  power  will  be  equal  to  the  momentum  of  the 
weight  ; and  their  velocities  will  be  in  reciprocal  proportion 
to  each  other.  For  the  weight  and  power  will  describe  cir- 
cles whose  radii  are  the  distances  cd,  ce  and  since  the  cir- 
cumferences or  spaces  described,  are  as  the  radii,  and  also 
as  the  velocities,  therefore  the  velocities  are  as  the  radii  cd, 
ce  ; and  the  momenta,  which  are  as  the  masses  and  velocities, 
are  as  the  masses  and  radii  ; that  is,  as  p X ce  and  w -X  cd, 
which  are  equal  by  cor.  3. 

' 173.  Corol.  5.  In  a straight  lever,  kept  in  equilibrio  by  a 
weight  and  power  acting  perpendicularly  ; then,  of  these 
three,  the  power,  weight,  and  pressure  on  the  prop,  any  one 
is  as  the  distance  of  the  other  two. 


B 


c 


1) 


A 

A 


©]5 


;v 


174.  Corol.  G.  If 
several  weights,  p,  q,  n, 
s,  act  on  a straight  le- 
ver, and  keep  it  in  equi- 
librio ; then  the  sum  of 
the  products  on  one  side 
of  the  prop  will  be 
equal  to  the  sum  on  the 

other  side,  made  by  multiplying  each  weight  by  its  distance  ; 
namely, 

P X AC  + Q X BC  = R X DC  + S X EC. 

For,  the  effect  of  each  weight  to  turn  the  lever,  is  as  the 
weight  multiplied  by  its  distance  ; and  in  the  case  of  an 
equilibrium,  the  sums  of  the  effects,  or  of  the  products  on  both 
sides  are  equal. 


175.  Corol.  7.  Because,  when 
t\yo  weights  q and  r are  in 
equilibrio,  $ : R : : cd  : cb  ; 

B C 

D 

: A 

therefore,  by  composition,  q -f  r : 
and,  o -j-  r 

@q 

Q : : bd  : cd, 
: r : : bd  : cb. 

R 

That 


MECHANICS. 


158 

That  is,  the  sum  of  the  weights  is  to  either  of  them,  as  the 
3um  of  their  distances  is  to  the  distance  of  the  other. 

SCHOLIUM. 

176.  On  the  foregoing  prin- 
ciples depends  the  nature  of 
scales  and  beams,  for  weigh- 
ing all  sorts  of  goods.  For, 
if  the  weights  be  equal,  then 
will  the  distances  be  equal  al- 
so, which  gives  the  construc- 
tion of  the  common  scales, 
which  ought  to  have  these  pro- 
perties : 

1st.  That  the  points  of  suspension  of  the  scales  and  the 
centre  of  motion  of  the  beam,  a,  b,  c,  should  be  in  a straight 
line  : 2 d,  That  the  arms  ab,  bc,  be  of  an  equal  length  : 3d, 
That  the  centre  of  gravity  be  in.  the  centre  of  motion  b,  or  a 
little  below  it : 4th,  That  they  be  in  equili ' rio  when  empty:  bth. 
That  there  be  as  little  friction  as  possible  at  the  centre  b.  A 
defect  in  any  of  these  properties,  makes  the  scales  either  im- 
perfect or  false.  But  it  often  happens  that  the  one  side  of  the 
beam  is  made  shorter  than  the  other,  and  the  defect  covered 
by  making  that  scale  the  heavier,  by  which  means  the  scales 
hang  in  equilibrio  when  empty  ; but  when  they  are  charged 
with  any  weights,  so  as  to  be  still  in  equilibrio,  those  weights 
are  not  equal  ; but  the  deceit  will  be  detected  by  changing  the 
weights  to  the  contrary  sides,  for  then  the  equilibrium  will 
be  immediately  destroyed. 

177.  To  find  the  true  weight  of  any  body  by  such  a false 
balance  : — First  w’eigh  the  body  in  one  scale,  and  afterwards 
weigh  it  in  the  other  ; then  the  mean  proportional  between 
these  two  weights,  will  be  the  true  weight  required.  For,  if 
any  body  b weigh  w pounds  or  ounces  in  the  scale  d,  and  only 
zv  pounds  or  ounces  in  the  scale  e : then  we  have  these  two 
equations,  namely,  ab  . b = bc  . w. 

and  bc  . b — ab  . w ; 

the  product  of  the  two  is  ab  . bc  b2  = ab  . bc  . wc  ; 
hence  then  - - - b2  — wb, 

and  - - - b2  = ^/wtr, 

the  mean  proportional,  which  is  the  true  weight  of  the  body  b . 

178.  The  Roman  Statera,  or  Steelyard,  is  also  a lever,  but 
of  unequal  brachia  or  arms,  so  contrived,  that  one  weight 
only  may  serve  to  weigh  a great  many,  by  sliding  it  back- 
ward 


THE  WHEEL  AND  AXLE.  150- 

ward  and  forward,  to  different  distances,  on  the  longer  arm 
of  the  lever  ; and  it  is  thus  constructed  : 


Let  ab  be  the  steelyard,  and  c its  centre  of  motioD,  whence 
the  divisions  must  commence  if  the  two  arms  just  balance 
each  other  : if  not,  slide  the  constant  moveable  weight  t 
along  from  b towards  c,  till  it  just  balance  the  other  end 
without  a weight,  and  there  make  a notch  in  the  beam, 
marking  it  with  a cipher  0.  Then  hang  on  at  a a weight  w 
equal  to  i,  and  slide  i back  towards  b till  they  balance  each 
other  ; there  notch  the  beam,  and  mark  it  with  1.  Then 
make  the  weight  w double  of  i,  and  sliding  i back  to  balance 
it,  there  mark  it  with  2.  Do  the  same  at  3,  4,  5,  &c.  by 
making  w equal  to  3,  4,  5,  &c.  times  i ; and  the  beam  is 
finished.  Then  to  find  the  weight  of  any  body  b by  the 
steelyard  ; take  off-  the  weight  w,  and  hang  on  the  body  b 
at  a ; then  slide  the  weight  i backward  and  forward  till  it 
just  balance  the  body  b,  which  suppose  to  be  at  the  number 
5 ; then  is  6 equal  to  5 times  the 'weight  of  i So,  if  i be  one 
pound,  then  b is  5 pounds  ; but  if  i be  2 pounds,  then  b is  IQ 
founds  ; and  so  on. 

OF  THE  WHEEL  AND  AXLE. 

PROPOSITION  XXXII. 

b79.  In  the  Wheel-and-Axle ; the  Weight  and  Power  will  be 
in  Equilibria , when  the  Power  p is  to  the  Weight  w,  Reci- 
procally as  the  Radii  of  the  Circles  where  they  act  ; that  is , 
»s  the  Radius  of  the  Axle  Ca,  where  the  Weight  hangs,  to 
the  Radius  of  the  Wheel  ce,  where  the  Power  acts.  Thai 
is,  p : wr  : : ca  : cb. 

HSR-E  the  cord,  by  which  the  pewer  a?  acts,  goes  about 

the 


MECHANICS. 


1.60 


the  circumference  of  the  wheel,  while 
that  of  the  weight  w goes  round  its 
axle,  or  another  smaller  wheel,  attach- 
ed to  the  larger,  and  having  the  same 
axis  or  centre  c.  So  that  ba  is  a lever 
moveable  about  the  point  c,  the  power 
I-  acting  always  at  the  distance  bc, 
and  the  weight  w at  the  distance  ca  ; 
therefore  p : w : : ca  : cb. 

ICO  Corol.  1.  If  the  wheel  be  put 
in  motion  ; then,  the  spaces  moved 
being  as  the  circumferences,  or  as  the  radii,  the  velocity  of 
w will  be  to  the  velocity  of  p,  as  ca  to  cb  ; that  is,  the 
weight  is  moved  as  much  slower,  as  it  is  heavier  than  the 
power  ; so  that  what  is  gained  in  power,  is  lost  in  time. 
And  this  is  the  universal  property  of  all  m ichine9  and  en- 
gines. 

181.  Corol.  2.  If  the  power  do  not  act  at  right  angles 
to  the  radius  cb,  but  obliquely  ; draw  cd  perpendicular  to 
the  direction  of  the  power  ; then,  by  the  nature  of  the  lever, 
s : w : : c a : cd. 


182.  To  this  power  be- 
long all  turning  or  wheel 
machines,  of  different  radii. 

Thus,  in  the  roller  turning 
on  the  a\is  or  spindle  ce, 
by  the  handle  cbd  ; the 
power  applied  at  b is  to 
the  weight  w on  the  roller 
as  the  radius  of  the  roller 
is  to  the  radius  cb  of  the 
handle. 

183.  And  the  same  for  all  cranes,  capstans,  windlasses,  and 
such  like  ; the  power  being  to  the  weight,  always  as  the  ra- 
dius or  lever  at  which  the  weight  acts,  to  that  at  which  the 
power  acts  ; so  that  they  are  always  in  the  reciprocal  ratio 
of  their  velocities.  And  to  the  same  principle  may  be  re- 
ferred the  gimblet  and  auger  for  boring  holes. 

181.  But  all  this,  however,  is  on  supposition  that  the  ropes 
or  cords,  sustaining  the  weights,  are  of  no  sensible  thickness. 
For,  if  the  thickness  be  considerable,  or  if  there  be  several 
folds  of  them,  over  one  another,  on  the  roller  or  barrel  ; then 
we  must  measure  to  the  middle  of  the  outermost  rope,  for 

the 


THE  WHEEL  AND  AXLE. 


161 


turn  the  wheel  q,  and  this  turn  the  small  wheel  or  axle  r, 
and  this  turn  the  wheel  s,  and  this  turn  the  axle  t,  and  this 
turn  the  wheel  v,  and  this  turn  the  axle  x,  which  raises  the 
weight  w ; then  p : w : : cb  . de  . fg  : ac  . bd  . ef.  And 
in  the  same  proportion  is  the  velocity  of  w slower  than  that 
of  p.  Thus,  if  each  wheel  be  to  its  axle,  as  10  to  1 ; then 
p : w : : l3  : 10*  or  as  1 to  1000.  So  that  a power  of  one 
pound  will  balance  a weight  of  1000  pounds  ; but  then, 
when  put  in  motion,  the  power  will  move  1000  times  faster 
than  the  weight. 

. Vol.  II.  22  OF 


the  radius  of  the  roller  ; or,  to  the  radius  of  the  roller  we 
must  add  half  the  thickness  of  the  cord,  when  there  is  but 
one  fold. 

185.  The  wheel-and-axle  has  a great  advantage  over  the 
simple  lever,  in  point  of  convenience.  For  a weight  can  be 
raised  but  a little  way  by  the  lever  ; whereas,  by  the  continual 
turning  of  the  wheel  and  roller,  the  weight  may  be  raised  to 
any  height,  or  from  any  depth. 

186.  By  increasing  the  number  of  wheels  too,  the  power 
may  be  multiplied  to  any  extent,  making  always  the  less 
wheels  to  turn  greater  ones,  as  far  as  we  please  ; and  this  is 
commonly  called  Tooth  and  Pinion  Work,  the  teeth  of  one 
circumference  working  in  the  rounds  or  pinions  of  another, 
to  turn  the  wheel.  And  then,  in  case  of  an  equilibrium,  the 
power  is  to  the  weight,  as  the  continual  product  of  the  radii 
of  all  the  axles,  to  that  of  all  the  wheels.  So,  if  the  power  p 


1(32 


MECHANICS. 


OF  THE  PULLEY. 


187.  A Pulley  is  a small  wheel,  commonly  made  of  wood 
or  brass,  which  turns  about  an  iron  axis  passing  through  the 
centre,  and  fixed  in  a block,  by  means  of  a cord  passed  round 
its  circumference,  which  serves  to  draw  up  any  weight. 
The  pulley  is  either  single,  or  combined  together,  to  increase 
the  power.  It  is  also  either  fixed  or  moveable,  according  as 
it  is  fixed  to  on»e  place,  or  moves  up  and  down  with  the. 
weight  and  power. 

PROPOSITION  XXXIII. 


188.  If  a Power  sustain  a Weight  by  means  of  a Fixed 
Pulley  : the  Power  and  Weight  are  Equal. 


For,  through  the  centre  c of  the  pulley 
draw  the  horizontal  diameter  ab  : then 
will  ab  represent  a lever  of  the  first  kind, 
its  prop  being  the  fixed  centre  c ; from 
which  the  points  a and  b,  where  the 
power  and  weight  act,  being  equally 
distant,  the  power  p is  consequently  equal 
to  the  weight  w. 

189  Corol.  Hence,  if  the  pulley  be  put 
in  motion,  the  power  p will  descend  as 
fast  as  the  weight  w ascends.  So  that 
the  power  is  not  increased  by  the  use  of 
the  fixed  pulley,  even  though  the  rope  go  over  several  of 
them.  It  is,  however,  of  great  service  in  the  raising  of 
weights,  both  by  changing  the  direction  of  the  force,  for  the 
convenience  of  acting,  and  by  enabling  a person  to  raise  a 
weight  to  any  height  without  moving  from  his  place,  and 
also  by  permitting  a great  many  persons  at  once  to  exert 
their  force  on  the  rope  at  p,  which  they  could  not  do  to  the 
weight  itself  ; as  is  evident  in  raising  the  hammer  or  weight 
of  a pile-driver,  as  well  as  on  many  other  occasions. 


PROPOSITION  XXXIV. 

1.90.  If  a Power  sustain  a Weight  by  means  of  One  Moveable 
Pulley  ; the  Power  is  but  Half  the  Weight. 

For,  here  ab  may  be  considered  as  a lever  of  the  second 

kind- 


THE  INCLINED  PLANE. 


163 


kind,  the  power  acting  at  a, 
the  weight  at  c,  and  the  prop 
or  fixed  point  at  b ; and  be- 
cause p : w : : cb  : ab, 
and  cb  = Aab,  therefore  p 
= }w,  or  w = 2p. 

3 91.  Cord.  1.  Hence  it  is 
evident,  that  when  the  pul- 
ley is  put  in  motion,  the  ve- 
locity of  the  power  will  be 
double  the  velocity  of  the 
weight,  as  the  point  p moves 
twice  as  fast  as  the  point  c and  weight  w rises.  It  is  also  evi- 
dent. that  the  fixed  pulley  f makes  no  difference  in  the  power 
p,  but  is  only  used  to  change  the  direction  of  it,  from  upwards 
to  downwards. 

192.  Corol.  2.  Hence  we  may  estimate  the  effect  of  a combi- 
nation of  any  number  of  fixed  and  moveable  pulleys  ; by 
which  we  shall  find  that  every  cord  going  over  a moveable 
pulley  always  adds  2 to  the  powers  ; since  each  moveable  pul- 
ley’s rope  bears  an  equal  share  of  the  weight ; while  each  rope 
that  is  fixed  to  a pulley,  only  increases  the  power  by  unity. 


to- \~w-\-w 


Here  p = 


Here  p = 


w. 


OF  THE  INCLINED  PLANE. 

193.  THE  Inclined  Plane,  is  a plane  inclined  to  the 
horizon,  or  making  an  angle  with  it.  It  is  often  reckoned  one 
of  the  simple  mechanic  powers  ; and  the  double  inclined  plane 
makes  the  wedge.  It  is  employed  to  advantantage  in  raising 
heavy  bodies  in  certain  situations,  diminishing  their  weights  by 
laying  them  on  the  inclined  planes. 


PROPOSITION 


164 


MECHANICS. 


PROPOSITION  XXXV. 

194.  The  Power  gained  by  the  Inclined  Plane,  is  in  Proportion 
as  the  Length  of  the  Plane  is  to  its  Height.  That  is,  when  a 
Weight  w is  sustained  on  an  'Inclined  Plane  ; bc,  by  a Power 
p acting  in  the  Direction  dw,  parallel  to  the  Plane  ; then  the 
Weight  w,  is  in  proportion  to  the  Power  p,  as  the  Length  of' 
the  Plane  is  to  its  Height ; that  is,  w : p : : bc  : ab. 

For,  draw  ae  perp.  to 
the  plane  bc,  or  to  dw. 

Then  we  are  to  consider 
that  the  body  w is  sustained 
by  three  forces,  viz.  1st,  its 
own  weight  or  the  force  of 
gravity,  acting  perp.  to  ac,  or  parallel  to  ba  ; 2d,  by  the 
power  p,  acting  in  the  direction  wd,  parallel  to  bc.  or  be  . 
and  .3dly,  by  the  re-action  of  the  plane,  perp.  to  its  face,  or 
parallel  to  the  line  ea.  But  when  a body  is  kept  in  equili- 
brio  by  the  action  of  three  forces,  it  has  been  proved,  that 
the  intensities  of  these  forces  are  proportional  to  the  sides 
of  the  triangle  aee  made  by  lines  drawn  in  the  directions 
of  their  actions  ; therefore  those  forces  are  to  one  another  as 
the  three  lines  - - - - ab,  be,  ae  ; that  is, 

the  weight  of  the  body  w is  as  the  line  ab, 
the  power  p is  as  the  line  - - be, 

and  the  pressure  on  the  plane  as  the  line  ae. 

But  the  two  triangles  abe,  abc  are  equiangular,  and  have 

therefore  their  like  sides  proportional  ; that  is, 

the  three  lines  ....  ab,  be,  ae, 

are  to  each  other  respectively  as  the  three  bc,  ab,  ac, 

or  also  as  the  three  ...  bc,  ae,  ce, 

which  therefore  are  as  the  three  forces  w,  p,  p, 

where  p denotes  the  pressure  on  the  plane.  That  is,  w : p . . 

bc  : ab,  or  the  weight  is  to  the  power,  as  the  length  of  the 

plane  is  to  its  height. 

See  more  on  the  Inclined  Plane,  at  p.  144,  &c. 

195.  Scholium.  The  Inclined  plane  comes  into  use  in  some 
situations  in  which  the  other  mechanical  powers  cannot  be 
conveniently  applied,  or  in  combination  with  them.  As,  in 
sliding  heavy  weights  either  up  or  down  a plank  or  other  plane 
laid  sloping  : or  letting  large  casks  down  into  a cellar,  or 
drawing  them  out  of  it.  Also,  in  removing  earth  from  a lower 
situation  to  a higher  by  means  of  wheel-barrows,  or  otherwise, 
as  in  making  fortifications,  &c.  ; inclined  planes,  made  of 
boards,  laid  aslope,  serve  for  the  barrows  to  run  upon. 


Of 


THE  WEDGE. 


165 


Of  all  the  various  directions  of  drawing  bodies  up  an  in- 
clined plane,  or  sustaining  them  on  it,  the  most  favourable 
is  where  it  is  parallel  to  the  plane  bc,  and  passing  through 
the  centre  of  the  weight ; a direction  which  is  easily  given 
to  it,  by  fixing  a pulley  at  d,  so  that  a cord  passing  over  it, 
and  fixed  to  the  weight,  may  act  or  draw  parallel  to  the  plane. 
In  every  other  position,  it  would  require  a greater  power  to 
support  the  body  on  the  plane,  or  to  draw  it  up.  For  if  one 
end  of  the  line  be  fixed  at  w,  and  the  other  end  inclined  down 
towards  b,  below  the  direction  wd,  the  body  would  be  drawn 
down  against  the  plane,  and  the  power  must  be  increased  in 
proportion  to  the  greater  difficulty  of  the  traction.  And,  on 
the  other  hand,  if  the  line  were  carried  above  the  direction 
of  the  plane,  the  power  must  be  also  increased  ; but-here  only 
in  proportion  as  it  endeavours  to  lift  the  body  off  the  plane. 

If  the  length  bc  of  the  plane  be  equal  to  any  number  of 
times  its  perp.  height  ab,  as  suppose  3 times  ; Ihen  a power 
p of  1 pound  hanging  freely,  will  balance  a weight  w of  3 
pounds,  laid  on  the  plane  : and  a power  p of  2 pounds,  will 
balance  a weight  w of  6 pounds  ; and  so  on,  always  3 times 
as  much.  But  then  if  they  be  set  a-moving,  the  perp.  descent 
of  the  power  p,  will  be  equal  to  3 times  as  much  as  the  perp. 
ascent  of  the  weight  w.  For,  though  the  weight  w ascends 
up  the  direction  of  the  oblique  plane,  bc,  just  as  fast  as  the 
power  p descends  perpendicularly,  yet  the  weight  rises 
only  the  perp.  height  ab,  while  it  ascends  up  the  whole 
length  of  the  plane  bc,  which  is  3 times  as  much  ; that  is, 
for  every  foot  of  the  perp.  rise,  of  the  weight,  it  ascends  3 feet 
up  in  the  direction  of  the  plane,  and  the  power  p descends  as 
much,  or  3 feet. 


OF  THE  WEDGE. 


196.  THE  Wedge  is  a piece  of 
wood  or  metal,  in  form  of  half  a rec- 
tangular prism,  af  or  eg  is  the 
breadth  of  its  back  ; ce  its  height ; 
gc,  bc  its  sides  : and  its  end  gbc  is 
composed  of  two  equal  inclined 
planes  c,ce,  ecf. 


PROPOSITION 


166 


MECHANICS. 


PROPOSITION  XXXVI. 

197.  When  a Wedge  is  in  Equilibria  ; the  Power  acting  a gains 
the  Back , is  to  the  Force  acting  P erpendicvlarly  against  either 
Side,  as  the  Breadth  of  the  Back  ab,  is  to  the  Length  of  the 
Side  ac  or  bc. 


For,  any  three  forces,  which  sustain  one 
another  in  equilibrio,  are  as  the  correspond- 
ing sides  of  a triangle  drawn  perpendicular 
to  the  directions  in  which  they  act.  But 
ab  is  perp.  to  the  force  acting  on  the  back, 
to  urge  the  wedge  forward  ; and  the  sides 
ac,  bc  are  perp.  to  the  forces  acting  on 
them  ; therefore  the  three  forces  are  as  ab, 
ac,  BC. 

198.  Corol.  The  force  on  the  back,  / ab, 

Its  effect  in  direct,  perp.  to  ac,  j ac, 

And  its  effect  parallel  to  ab  ; S dc, 

are  as  the  ihree  lines  ( which  are  per.  to  them. 

And  therefore  the  thinner  a wedge  is,  the  greater  is  its 
effect  in  splitting  any  body,  or  in  overcoming  any  resistance 
against  the  sides  of  the  wedge. 


SCHOLIUM. 

199.  But  it  must  be  observed,  that  the  resistance,  or  the 
forces  above-mentioned,  respect  one  side  of  the  wedge  only. 
For  if  those  against  both  sides  be  taken  in.  then,  in  the  fore- 
going proportions,  we  must  take  only  half  the  back  ad,  or  else 
we  must  take  double  the  line  ac  or  dc. 

In  the  wedge,  the  friction  against  the  sides  is  very  great,  at 
least  equal  to  the  force  to  be  overcome,  because  the  wedge 
retains  any  position  to  which  it  is  driven  ; and  therefore  the 
resistance  is  double  by  the  friction.  But  then  the  wedge 
has  a great  advantage  over  all  the  other  powers,  arising  from 
the  force  of  percussion  or  blow  with  which  the  back  is  struck, 
which  is  a force  incomparably  greater  than  any  dead  weight 
or  pressure,  such  as  is  employed  in  other  machines.  And  ac: 
cordingly  wre  find  it  produces  effects  vastly  superior  to  those 
of  any  other  power  ; such  as  the  splitting  and  raising  the 
largest  and  hardest  rocks,  the  raising  and  lifting  the  largest 
ship,  by  driving  a wedge  below  it,  which  a man  can  do  by  the 
blow  of  a mallet  ; and  thus  it  appears  that  the  small  blow  of  a 
hammer,  on  the  back  of  a wedge,  is  incomparably  greater  than 
any  mere  pressure,  and  will  overcome  it. 


OF 


THE  SCREW. 


167 


OF  THE  SCREW. 

200.  THE  Screw  is  one  of  six  mechanical  powers, chiefly 
used  in  pressing  or  squeezing  bodies  close,  though  some- 
times also  in  raising  weights. 

The  screw  is  a spiral  thread  or  groove  cut  round  a cylin- 
der, and  every  where  making  the  same  angle  with  the  length 
of  it.  So  that  if  the  surface  of  the  cylinder,  with  this  spiral 
thread  on  it,  were  unfolded  and  stretched  into  a plane,  the 
spiral  thread  would  form  a straight  inclined  plane,  whose 
length  would  be  to  its  height,  as  the  circumference  of  the 
cylinder,  is  to  the  distance  between  two  threads  of  the 
screw:  as  is  evident  by  considering  that,  in  making  one 
round,  the  spiral  rises  along  the  cylinder  the  distance  between 
the  two  threads. 

PROPOSITION  XXXVII. 

201.  The  Force  of  a Power  applied  to  turn  a Screw  round,  is  to 
the  Force  with  which  it  presses  upward  or  downward,  setting 
aside  the  Friction,  as  the  Distance  between  two  Threads,  is 
to  the  Circumference  where  the  Power  is  applied. 

The  screw  being  an  inclined  plane,  or  half  wedge,  whose 
height  is  the  distance  between  two  threads,  and  its  base  the 
circumference  of  the  screw  ; and  the  force  in  the  horizontal 
direction,  being  to  that  in  the  vertical  one,  as  the  lines  per- 
pendicular to  them,  namely,  as  the  height  of  the  plane,  or 
distance  of  the  two  threads,  is  to  the  base  of  the  plane,  or 
circumference  of  the  screw  ; therefore  the  power  is  to  the 
pressure,  as  the  distance  of  two  threads  is  to  that  circumfer- 
ence. But,  by  means  of  a handle  or  lever,  the  gain  in  power 
is  increased  in  the  proportion  of  the  radius  of  the  screw  to 
the  radius  of  the  power,  or  length  of  the  handle,  or  as  their 
circumferences.  Therefore,  finally,  the  power  is  to  the  pres- 
sure, a?  the  distance  of  the  threads,  is  to  the  circumference 
described  by  the  power. 

202.  Corol.  When  the  screw  is  put  in  motion  ; then  the 
power  is  to  the  weight  which  would  keep  it  in  equilibrio,  as 
the  velocity  of  the  latter  is  to  that  of  the  former  ; and  hence 
their  two  momenta  are  equal,  which  are  produced  by  mul- 
tipl  ying  each  weight  or  power  by  its  own  velocity.  So  that 
this  is  a general  property  in  all  the  mechanical  powers, 
namely,  that  the  momentum  of  a power  is  equal  to  that  of 
the  weight  which  would  balance  it  in  equilibrio  ; or  that 
each  of  them  is  reciprocally  proportional  to  its  velocity. 

SCHOLIUM. 


ib8 


MECHANICS. 


SCHOLIUM. 


203.  Hence  we  can  easily 
compute  the  force  of  any  ma- 
chine turned  by  a screw.  Let 
the  annexed  figure  represent  a 
press  driven  by  a screw,  whose 
threads  are  each  a quarter  of 
an  inch  asunder  ; and  let  the 
screw  be  turned  by  a handle 
of  4 feet  longt  from  a to  b ; 
then,  if  the  natural  force  of 
a man,  by  which  he  can  lift, 
pull,  or  draw,  be  150  pounds  ; and  it  be  required  to  deter 
mine  with  what  force  the  screw  will  press  on  the  board  at  d, 
when  the  man  turns  the  handle  at  a and  b,  with  his  whole 
force.  I hen  the  diameter  ab  of  the  power  being  4 feet,  or 
48  inches,  its  circumference  is  48  X 3-1416  or  1504  nearly; 
and  the  distance  of  the  threads  being  } of  an  inch  ; there- 
fore the  power  is  to  the  pressure  as  1 to  603^  : but  the 
power  is  equal  to  1501b  ; theref.  as  1 : 603i  : : 150  : 90480  ; 
and  consequently  the  pressure  at  d is  equal  to  a weight  of 
90480  pounds,  independent  of  friction. 


204.  Again,  if  the  end- 
less screw  ab  be  turned  by 
a handle  ac  of  20  inches, 
the  threads  of  the  screw 
being  distant  half  an  inch 
each  ; and  the  screw  turns 
a toothed  wheel  e,  whose 
pinion  l turns  another 
wheel  f,  and  the  pinion  m 
of  this  another  wheel  g,  to 
the  pinion  or  barrel  of 
which  is  hung  a weight  w ; 
it  is  required  to  determine 
what  weight  the  man  will 
be  able  to  raise,  working  at 
the  handle  c ; supposing  the 
diameters  of  the  wheels  to 
be  18  inches,  and  those  of 
the  pinions  and  barrel  2 
inches  ; the  teeth  and  pin- 
ions being  all  of  a size. 


Here 


CENTRE  OF  GRAVITY, 


169 


Here  20  X 3-1416  X 2 = 125-664,  is  the  circumference  of 
the  power. 

And  125-664  to  4,  or  251-328  to  1,  is  the  force  of  the  screw 
alone. 

Also,  18  to  2,  or  9 to  1,  being  the  proportion  of  the 
wheels  to  the  pinions  ; and  as  there  are  three  of  them, 
therefore  9 3 to  l3,  or  729  to  1,  is  the  power  gained  by  the 
wheels. 

Consequently  251-328  X 729,  to  1,  or  1832181  to  1 nearly, 
is  the  ratio  of  the  power  to  the  weight,  arising  from  the  advan- 
tage both  of  the  screw  and  the  wheels. 

But  the  power  is  1501b  ; therefore  150  X 183218^,  or 
27482716  pounds,  is  the  weight  the  man  can  sustain,  which  is 
equal  to  12269  tons  weight. 

But  the  power  has  to  overcome,  not  only  the  weight,  but 
also  the  friction  of  the  screw,  which  is  very  great,  in  some 
cases  equal  to  the  weight  itself,  since  it  is  sometimes  sufficient 
to  sustain  the  weight,  when  the  power  is  taken  off. 


ON  THE  CENTRE  OF  GRAVITY. 


205.  THE  Centre  of  Gravity  of  a body,  is  a certain 
point  within  it,  on  which  the  body  being  freely  suspended,  it 
will  rest  in  any  position  ; and  it  will  always  descend  to  the 
i lowest  place  to  which  it  can  get,  in  other  positions. 


PROPOSITION  XXXVIII, 

206.  If  a Perpendicular  to  the  Horizon,  from  the  centre  of 
Gravity  of  any  body,  fall  within  the  Base  of  the  Body,  it  will 
rest  in  that  Position  ; but  if  the  Perpendicular  fall  without 
the  Base,  the  Body  will  not  rest  in  that  Position,  but  will 
tumble  down.  . 

For,  if  cb,  be  the  perp. 
from  the  centre  of  gravity  c, 
within  the  base  : then  the 
body  cannot  fall  over  towards 
a ; because,  in  turning  on  the 
point  a,  the  centre  of  gravity 
g would  describe  an  arc  d 

which  would  rise  from  c to  e ; 
contrary  to  the  nature  of  that  centre,  which  only  rests  when 
m the  lowest  place.  For  the  same  reason,  the  body  will  not 
|tall  towards  d.  And  therefore  it  will  stand  in  that  position. 

! Vor,.  IT.  23  But 


m 


STATICS. 


But  if  the  perpendicular  fall  without  the  base,  as  cb  ; theii 
the  body  will  tumble  over  on  that  side  : because  in  turning  on 
the  point  a,  the  centre  c descends  by  describing  the  descend- 
ing arc  ce. 

207  Carol.  1.  If  a perpendicular,  drawn  from  the  centre 
of  gravity,  fall  just  on  the  extremity  of  the  base  , the  body 
may  stand  ; but  any  the  least  force  will  cause  it  to  fall  that 
way.  And  the  nearer  the  perpendicular  is  to  any  side,  or 
the  narrower  the  base  is,  the  easier  it  will  be  made  to  fall  or 
be  pushed  over  that  way  ; because  the  centre  of  gravity  has 
the  less  height  to  rise  : which  is  the  reason  that  a globe  is 
is  made  to  roll  on  a smooth  plane  by  any  the  least  force. 
But  the  nearer  the  perpendicular  is  to  the  middle  of  the 
base  or  the  broader  the  base  is,  the  firmer  the  body  stands. 

2QP.  Carol  2.  Hence  if  the  centre  of  gravity  of  a body 
be  supported,  the  whole  body  is  supported.  And  the  place 
of  the  centre  of  gravity  must  be  accounted  the  place  of  the 
body  : for  into  that  point  the  whole  matter  of  the  body  may 
be  supposed  to  be  collected,  and  therefore  all  the  force  alse 
with  which  it  endeavours  to  descend. 

209.  Carol.  3.  From  the  property  which 
the  centre  of  gravity  has,  of  always  des- 
cending to  the  lowest  point,  is  derived  an 
easy  mechanical  method  of  finding  that 
centre. 

Thus,  if  the  body  be  hung  up  by  any 
point  a,  and  a plumb  line  ab  be  hung  by 
the  same  point,  it  will  pass  through  the  cen- 
tre of  gravity  ; because  that  centre  is  not  in 
the  lowest  point  till  it  fall  in  the  plumb 
line.  Mark  the  line  ab  on  it.  Then  hang 
the  body  up  by  any  other  point  d,  with  a 
plumb  line  df.,  which  will  also  pass  through 
the  centre  of  gravity,  for  the  same  reason 
a?  before  ; and  therefore  that  centre  must 
be  at  c where  the  two  plumb  lines  cross 
each  other. 


— G 


210.  Or,  if  the  body  be  suspended  by 
two  or  more  cords  gf,  gh,  &c.  then  a 
plumb  line  from  the  point  g will  cut  the 
body  in  its  centre  of  gravity  e. 


211.  Like- 


CENTRE  OP  GRAVITY. 


171 


211.  Likewise,  because  a body  rests  when  its  centre  of 
gravity  is  supported,  but  not  else  ; we  hence  derive  another 
easy  method  of  boding  that  centre  mechanically.  For,  if 
the  body  be  laid  on  the  edge  of  a prism,  or  over  one  side  of 
a table,  and  moved  backward  and  forward  till  it  rest,  or  ba- 
lance itself ; then  is  the  centre  of  gravity  just  over  the  line  of 
the  edge.  And  if  the  body  be  then  shifted  into  another  po- 
sition, and  balanced  on  the  edge  again,  this  line  will  also 
pass  by  the  centre  of  gravity  ; and  consequently  the  inter- 
section of  the  two  will  give  the  centre  itself. 

PROPOSITION  XXXIX. 

212.  The  common  Centre  of  Gravity  c of  any  two  Bodies  a,  e, 
divides  the  Line  joining  their  Centres , into  two  Parts,  which 
are  Reciprocally  as  the  Bodies. 

That  is,  ac  : bc  : : b : a. 

For,  if  the  centre  of  gravity  c be  supported,  the  two 

bodies  a and  b will  be  supported,  

and  will  rest  in  equilibrio  But  <5  5? 

by  the  nature  of  the  lever,  when  ^ 

two  bodies  are  in  equilibrio  about  a fixed  point  c,  they  are 
reciprocally  as  their  distances  from  that  point ; therefore 
a : b : : cb  : ca. 

213.  Corol.  1.  Hence  ab  : ac  : : a + b : b ; or,  the  whole 
distance  between  the  two  bodies,  is  to  the  distance  of  either 
of  them  from  the  common  centre,  as  the  sum  of  the  bodies  is 
to  the  other  body. 

214  Corol.  2.  Hence  also,  ca  . a = cb  . b ; or  the  two 
products  are  equal,  which  are  made  by  multiplying  each  body 
by  its  distance  from  the  centre  of  gravity. 

215.  Corol.  3.  As  the  centre  c is  pressed  with  a force  equal 
to  both  the  weights  a and  b,  while  the  points  a and  b are 
each  pressed  with  the  respective  weights  a and  b.  There- 
fore, if  the  two  bodies  be  both  united  in  their  common 
centre  c,  and  only  the  ends  a and  b of  the  line  ab  be  sup- 
ported, each  will  still  bear,  or  be  pressed  by  the  same  weights 
a and  b as  before.  So  that,  if  a weight  of  1001b  be  laid  on 
a bar  at  c,  supported  by  two  men  at  a and  b,  distant  from  c, 
the  one  4 feet,  and  the  other  6 feet  ; then  the  nearer  will 
bear  the  weight  of  601b,  and  the  farther  only  401b.  weight. 

216.  CoroL 


172 


STATICb. 


216.  Corol.  4.  Since  the 
effect  of  any  body  to  turn 
a lever  about  the  fixed 
point  c,  is  as  that  body  and 
as  its  distance  from  that  point  ; therefore,  if  c be  the  com- 
mon centre  of  gravity  of  all  the  bodies  a,  b,  d,  e,  f,  placed  in 
the  straight  line  af  ; then  is  ca  . a -f-  cb.b=cd.d  + 
ce  . e + cf  f ; or,  the  sum  of  the  products  on  one  side 
equal  to  the  sum  of  the  products  on  the  other,  made  by  mul- 
tiplying each  body  by  its  distance  from  that  centre.  And 
if  several  bodies  be  in  equilibrium  on  any  straight  lever,  then 
the  prop  is  in  the  centre  of  gravity. 

217.  Corol.  5.  And  though 
the  bodies  be  not  situated  in 
a straight  line,  but  scattered 
about  in  any  promiscuous  man- 
ner, the  same  property  as  in  the 
last  corollary  still  holds  true, 
if  perpendiculars  to  any  line 
whatever,  af  be  drawn  through 
the  several  bodies,  and  their  common  centre  of  gravity,  name- 
ly. that  ca  : a + cb  = cd  . d + ce  . e -)-  cf . f.  For  the 
bodies  have  the  same  effect  on  the  line  af,  to  turn  it  about  the 
point  c,  whether  they  are  placed  at  the  points  a,  b,  d,  e,f,  or 
in  any  part  of  the  perpendiculars  a a,  b b,  ad,  ec,  vf. 

PROPOSITION  XL. 

218.  If  there  be  three  or  more  Bodies,  and  if  a line  be  drawn 
from  any  one  Body  d to  the  Centre  of  Gravity  of  the  rest  c ; 
then  the  Common  Centre  of  Gravity  e of  all  the  Bodies,  divides 
the  line  cn  into  two  Parts  in  e,  which  are  Beciprocally  Pro- 
portional as  the  Body  d to  the  sum  of  all  the  other  Bodies 

That  is,  qe  : ed  : : d : a + b &c. 

For,  suppose  the  bodies  a and  e 
to  be  collected  into  the  common  ^ 
centre  of  gravity  c,  and  let  their  sum 
be  called  s.  Then,  by  the  last  prop. 
ce  : ed  : : d : s or  a -f  b ike, 

217.  Corol.  Hence  we  have  a method  of  finding  the  com- 
mon centre  of  gravity  of  any  number  of  bodies  ; namely',  by 
first  finding  the  centre  of  any  two  of  them,  then  the  centre 
of  that  centre  and  a third,  and  so  on  for  a fourth,  or  fifth, 

&c. 


P a 


d 


£ 


6 

Id  fc 


PROPOSITION 


CENTRE  OF  GRAVITY, 


173 


PROPOSITION  XLL 

220.  If  there  be  taken  any  Point  p,  in  the  Line  passing  through 
the  Centres  of  two  Bodies  ; then  the  sum  of  the  two  Products , 
of  each  Body  multiplied  by  its  Distance  from  that  Point,  is 
equal  to  the  Product  of  the  Sum  of  the  Bodies  multiplied  by 
the  Distance  of  their  Common  Centre  of  Gravity  c from  the 
same  Point  p. 


That  is,  pa  . a *f-  pb  . b = pc  . a -f  B. 

For,  by  the  38th,  ca  . a = cb  . b, 
that  is,  pa — pc  . a = pc — pb  . b ; q p ® p 

therefore  by  adding, 
pa  . a + pb  . b = pc  . a -j-  B. 

221.  Corol.  1.  Hence,  the  two  bodies  a and  b have  the 
same  force  to  turn  the  lever  about  the  point  p,  as  if  they 
were  both  placed  in  c their  common  centre  of  gravity. 

Or,  if  the  line,  with  the  bodies,  move  about  the  point  p ; 
the  sum  of  the  momenta  of  a and  b,  is  equal  to  the  mo- 
mentum of  the  sums,  orA-f  b placed  at  the  centre  c. 

222.  Corol.  2.  The  same  is  also  true  of  any  number  of 
bodies  whatever,  as  will  appear  by  cor.  4,  prop.  39,  namely, 
pa  . a -f-  pb  . b -f-  pd  . d &c.  = pc  . a + b -j-  d &c.  where 
p is  in  any  point  whatever  in  the  line  ac. 

And,  by  cor.  5,  prop.  39,  the  same  thing  is  true  when  the 
bodies  are  not  placed  in  that  line,  but  any  where  in  the  per- 
pendiculars passing  through  the  points  a,  b,  d,  &c.  ; namely, 
pa  . a + pb  . b -f-  pd  . n & c.  = pc  . a + b + d &c. 

223.  Corol.  3.  And  if  a plane  pass  through  the  point  p per- 
pendicular to  the  line  cp  ; then  the  distance  of  the  common 
centre  of  gravity  from  that  plane,  is 

pa  . a + pb  - b + pd  . d &c.  , . , , ,. 

pc  = ; , that  is,  equal  to  the  sum 

a + B + d he.  ’ H 

of  all  the  forces  divided  by  the  sum  of  all  the  bodies.  Or, 
if  a,  b,  d,  &c,  be  the  several  particles  of  one  mass  or  com- 
pound body  ; then  the  distance  of  the  centre  of  gravity  of  the 
body,  below  any  given  point  p,  is  equal  to  the  forces  of  all 
the  particles  divided  by  the  whole  mass  or  body,  that  is,  equal 
to  all  the  pa  a,  pb  . b,  pd  , d,  &c.  divided  by  the  body  or  sum 
of  particles  a,  b,  d,  &c. 


PROPOSITION 


STATICS. 


m 


PROPOSITION  XLU. 


224.  To  find  the  Centre  of  Gravity  of  any  Body,  or  of  any  Sys- 
tem of  Bodies. 


©15 

P a I 

G- 

1 

I 

IE 

f 

! h 

c 

e 1 
Id  I® 

Through  any  point  t>  draw 
a plane,  and  let  pa,  pb,  pd,  &c. 
be  the  distance  of  the  bodies 
a,  b,  d,  &c.  from  the  plane  ; 
then,  by  the  last  cor.  the  dis- 
tance of  the  common  centre  of 
gravity  from  the  plane,  will  be 

pa  . a pb  • b pd  d Sec. 

' 6 A + B D & c. 


225.'  Or,  if  b be  any  body,  and  qpr  any  plane  ; draw  fab 
&c.  perpendicular  to  qr,  and  through  a,  b,  &c.  draw  innu- 
merable sections  of  the  body  b parallel 
to  the  plane  qr.  Let  s denote  any  of 
these  sections,  and  d — pa,  or  pb,  &c. 
its  distance  from  the  plane  qr.  Then 
will  the  distance  of  the  centre  of  gra- 
vity of  the  body  from  the  plane  be 
sum  of  all  the  rf’s 

pc  — . And  if  the 


distance  be  thus  found  for  two  inter- 
secting planes,  they  will  give  the  point 
in  which  the  centre  is  placed. 


226.  But  the  distance  from  one  plane  is  sufficient  for  any 
regular  body,  because  it  is  evident  that  in  such  a figure,  the 
centre  of  gravity  is  in  the  axis,  or  line  passing  through  the 
centres  of  all  the  parallel  sections. 

Thus,  if  the  figure  be  a parallelogram,  or  a 
cylinder,  or  any  prism  whatever  ; then  the  axis 
or  line,  or  plane  ps,  which  bisects  all  the  sec- 
tions parallel  to  qr;  will  pass  through  the 
centre  of  gravity  of  all  those  sections,  and 
consequently  through  that  of  the  whole  figure 
c.  Then,  all  the  sections  s being  equal,  and 
the  body  b — ps  . s,  the  distance  of  the  centre 
will  be  pc  = 


A 

“D 

J D 

Tv 

J~r 

c 

s 

PA  .S  + PB  - S + S.C.  PA  -f-  PB  -f-  PD  StC. 


X s 


PA  -f  PB  4-  Sec. 
PS 

But 


CENTRE  OF  GRAVITY. 


lib 


But  pi  + p*  + &c  is  the  sum  of  an  arithmetical  pro- 
gression, beginning  at  0,  and  increasing  to  the  greatest  term 
rs,  the  number  of  the  terms  being  also  equal  to  ps  ; there- 
fore the  sum  pa  -j-  pb  + &c.  — ^ ps  . ps  ; and  consequently 

pc  = — - ‘ P-  = i ps  ; that  is,  the  centre  of  gravity  is  in 


the  middle  of  the  axis  of  any  figure  whose  parallel  sections 
are  equal. 


227.  In  other  figures,  whose  parallel  sections  are  not 
equal,  but  varying  according  to  some  general  law,  it  will  not 
be  easy  to  find  the  sum  of  all  the  pa  . s,  pb  . s',  pd  . s ',  &c. 
except  by  the  general  method  of  Fluxions  ; which  case 
therefore  will  be  best  reserved,  till  we  come  to  treat  of  that 
doctrine.  It  will  be  proper  however  to  add  here  some  ex- 
amples of  another  method  of  finding  the  centre  of  gravity  of  a 
triangle,  or  any  other  right-lined  plane  figure. 


PROPOSITION  XLIIT, 

228.  To  find  the  Centre  of  Gravity  of  a Triangle. 

From  any  two  of  the  angles  draw 
lines  ad,  ce,  t«>  bisect  the  opposite 
sides,  sa  will  their  intersection  g be 
the  centre  of  gravity  of  the  triangle. 

For,  because  ad  bisects  bc,  it  bi- 
sects also  all  its  parallels,  namely,  all 
the  parallel  sections  of  the  figure ; 
therefore  ad  passes  through  the  cen- 
tres of  gravity  of  all  the  parallel  sections  or  component  parts 
of  the  figure  ; and  consequently  the  centre  of  gravity  of  the 
whale  figure  lies  in  the  line  ad.  For  the  same  reason,  it  also 
lies  in  the  line  ce.  Consequently  it  is  in  their  common  point 
•f  intersection  g. 

229  Corol.  The  distance  of  the  point  g,  is  ac,  = § ad,  and 
cg  = | ce  : or  ag  = 2gd,  and  cg  = 2ge. 

For,  draw  bf  parallel  to  ad,  and  produce  ce  to  meet  it 
in  f.  Then  the  triangles  aeg,  eef  are  similar,  and  also 
equal,  because  ae  = be  ; consequently  ag  = bf.  But 
the  triangles  cdg,  cbf  are  also  equiangular,  and  cb  being 
= 2cd,  therefore  bf  = 2gd.  But  bf  is  also  = ag  ; 
consequently  ag  2gd,  ©r  §ad.  In  like  manner,  cg  = 2ge 
or  §ce  . 


PROPOSITION 


176 


STATICS. 


PROPOSITION  XLTV. 

230.  To  find  the  Centre  of  Gravity  of  a Trapezium. 

Divide  (he  trapezium  abcd  into 
two  triangles,  by  the  diagonal  ed,  and 
find  e,  f,  the  centres  of  gravity  of 
these  two  triangles  ; then  shall  the 
centre  of  gravity  of  the  trapezium  lie 
in  the  line  ef  connecting  them.  And 
therefore  if  ef  be  divided,  inG,  in  the 
alternate  ratio  of  the  two  triangles, 
namely,  eg  : ge  : : triangle  bcd  : triangle  abd,  then  g will  be 
the  centre  of  gravity  of  the  trapezium. 

231.  Or,  having  found  the  two  points  e,  f,  if  the  trape- 
zium be  divided  into  two  other  triangles  bac,  dac,  by  the  other 
diagonal  ac,  and  the  centres  of  gravity  h and  i of  these  two 
triangles  be  also  found  ; then  the  centre  of  gravity  of  the  tra- 
pezium will  also  lie  in  the  line  hi. 

So  that,  lying  in  both  the  lines,  ef,  hi,  it  must  necessarily 
lie  in  their  intersection  g. 

232.  And  thus  we  are  to  proceed  for  a figure  of  any 
greater  number  of  sides,  finding  the  centres  of  their  compo- 
nent triangles  and  trapeziums,  and  then  finding  the  com- 
mon centre  of  every  two  of  these,  till  they  be  all  reduced 
into  one  only. 

Of  the  use  of  the  place  of  the  centre  of  gravity,  and  the 
nature  of  forces,  the  following  practical  problems  are  added  ; 
viz.  to  find  the  force  of  a bank  of  the  earth  pressing  against  a 
wall  and  the  force  of  the  wall  to  support  it  ; also  the  push  of 
an  arch,  with  the  thickness  of  the  piers  necessary  to  support 
it  ; also  the  strength  and  stress  of  beams  and  bars  of  timber 
and  metal,  &c. 

PROPOSITION  XLV. 


A D 


A I BC 


233.  To  determine  the  Force  with  which  a Bank  of  Earth , or 
such  like,  presses  against  a Wall,  and  the  Dimensions  of  the 
Wall  necessary  to  Support  it. 

Let  acde  be  a vertical  section  of  a 
bank  of  earth  ; and  suppose,  that  if  it 
were  not  supported,  a triangular  part  of 
it,  as  abe,  would  slide  down,  leaving 
it  at  what  is  called  the  natural  slope  be  ; 
but  that,  by  means  of  a wall  aefg,  it 
is  supported,  and  kept  in  its  place. — It 
is  required  to  find  the  force  of  abe, 
to  slide  down . and  the  dimensions  of  the 
the  wall  aefg,  to  support  it. 


Let 


CENTRE  OF  GRAVITY. 


177 


Let  h be  the  centre  of  gravity  of  the  triangle  abe,  through 
which  draw  khi  parallel  to  the  slope  face  of  the  earth  be. 
Now  the  centre  of  gravity  h may  be  accounted  the  place  of 
the  triangle  abe,  or  the  point  into  which  it  is  all  collected. 
Draw.HL  parallel,  and  kp  perpendicular  to  ae,  also  kl  prep, 
to  ik  or  be.  Then  if  hl  represent  the  force  of  the  triangle 
abe  in  its  natural  direction  hl,  hk  will  denote  its  force  in 
its  direction  hk,  and  pk  the  same  force  in  the  direction  pk 
perpendicular  to  the  lever  ek,  on  which  it  acts.  Now  the 
three  triangles  eab,  hkl,  hkp  are  all  similar  ; therefore 
eb  : ea  : : (hl  : hk  : :)  w the  weight  of  the  triangle  eab  : 

JE  A 

— w,  which  will  be  the  force  of  the  triangle  in  the  direc- 

tion  hk  Then,  to  find  the  effect  of  this  force  in  the  direc- 
..  . ...  . ea  ea  . AB 

tion  pk,  it  will  be,  as  hk  : pk  : : eb  : ab  : : — -a>  : 

EB  EB2 

the  force  at  k,  in  direction  pk,  perpendicularly  on  the  lever 
ek,  which  is  equal  to  Iae.  But  f ae  . ab  is  the  area  of  the 
triangle  abe  ; and  if  m be  the  specific  gravity  of  the  earth, 
then  | ae  . ab  . m is  as  its  weight.  Therefore 

EA  • AB  E A 2 . ab-  . . . . 

— . Aae  . ab  = — m is  the  force  acting  at  k in 

f-b2  z 2eb  55 

direction  pk.  And  the  effect  of  this  pressure  to  overturn 
the  wall,  is  also  as  the  length  of  the  lever  ke  or  ^ae*  : con- 


* The  principle  now  employed  in  the  solution  of  this  4-5th  piop, 
is  a little  different  from  that  formerly  used  ; viz.  by  considering  the 
triangle  of  earth  abe  as  acting  by  lines  ik,  &cc-  parallel  to  the  face 
of  the  slope  be,  instead  of  acting  in  directions  parallel  to  the  horizon 
ab  ; an  alteration  which  gives  the  length  of  the  lever  ek,  only  the 
half  ot  what  it  was  in  the  former  way,  viz.  ek  = |ae  instead  of f ae : 
but  every  thing  else  remaining  the  same  as  before,  indeed  this  prob- 
lem has  formerly  been  treated  on  a variety  of  different  hypotheses,  by 
Mr.  Muller,  &C.  in  this  country,  and  by  many  French  and  other  au- 
thors in  other  countries-  And  this  has  been  chiefly  owing  to  the  un- 
certain way  in  which  loose  earth  may  be  supposed  to  act  in  such  a 
j ease  ; which  on  account  of  its  various  circumstances  of  tenacity,  fric- 
tion, 8tc.  will  not  perhaps  admit  of  a strict  mechanical  certainty.  On 
f these  accounts  it  seems  probable  that  it  is  to  good  experiments  only, 
made  on  different  kinds  of  earth  and  walls,  that  we  may  probably 
hope  for  a just  and  satisfactory  solution  of  the  problem. 

The  above  solution  is  given  only  in  the  most  simph  case  of  the 
problem*  But  the  same  principle  may  easily  be  extended  to  any 
other  case  that  may  be  required,  either  in  theory  or  practice,  either 
with  walls  or  banks  of  earth  of  different  figures,  and  in  different 
situations. 


Vot.  IT. 


24 


sequently 


178 


STATICS. 


sequently  its  effect  is 


EA3  . A B “ 


6e  b1 


m,  for  the  perpendicular  force 


against  K.to  overset  the  wall  aefg.  Which  must  be  balanced 
b\  (lie  counter  resistance  of  the  wall,  in  order  that  it  may  at 
least  be  supported. 


Now,  if  m be  the  centre  of  gravity  of  the  wall,  into  which 
its  whole  matter  may  be  supposed  to  be  collected,  and  acting 
in  the  direction  mnw,  its  effect  will  be  the  same  as  if  a weight 
w were  suspended  from  the  point  n of  the  lever  fn.  Hence, 
if  a be  put  for  the  area  of  the  wall  aefg,  and  n its  specific 
gravity  ; then  a . n will  be  equal  to  the  weight  w,  and  a . 
n . fn  its  effect  on  the  lever  to  prevent  it  from  turning  about 
the  point  f.  And  as  this  effort  must  be  equal  to  that  of  the 
triangle  of  earth,  that  it  may  just  support  it,  which  was 


before  found  equal  to 


ea 3 . AB2 
6eb2 


therefore  a . n . fn 


AE3  . AB- 


— — ' ■ -m,  in  case  of  an  equilibrium. 


234.  But  now,  both  the  breadth  of  the  wall  fe,  and  the  l 
lever  fn,  or  place  of  the  centre  of  gravity  m,  will  depend  on 
the  figure  of  the  wall  If  the  wall  be  rectangular,  or  as 
broad  at  top  as  bottom  ; then  fn  = -i  fe,  and  the  area  a = 
ae  . fe  ; consequently  the  effort  of  the  wall  a . n . fn  is  = 

■Ife2  ; ae  . n ; which  must  be  — in,  the  effort  of 

6eb2 

the  earth  And  the  resolution  of  this  equation  gives  the 

breadth  of  the  wall  fe  = ~-B~-  A_  / T"  =Aqa/—  , drawing  aq 

eb  3n  on 

perp  to  fb.  So  that  the  breadth  of  the  wall  is  always  pro- 
portional to  the  prep  depth  aq  of  the  triangle  abe  But  the 
breadth  must  be  made  a little  more  than  the  above  value 
of  it.  that  it  may  be  more  than  a bare  balance  to  the  earth. - 
If  the  angle  of  the  slope  e be  45°,  as  it  is  nearly  in  most  cases: 

,,  ae  tn  m m . 

then  fe  ===  — ./ — = ae  a/  - — = Iae*/ — very  nearly. 

on  v 6 n 3 v n J 

235.  If  the  wall  be  of  brick,  its  specific  gravity  is  about  * 
2C00,  and  that  of  the  earth  about  1984  ; namely,  m to  n as  1S84 

to  2000  ; or  they  may  be  taken  as  equal  ; then  ^ - — 1 very 

nearly  ; and  hence  fe  — t47ae,  or  |ae  nearly.  That  is, 
whenever  a brick  rectangular  wall  is  made  to  support  earth, 
its  thickness  must  be  at  least  f or  of  its  height.  But  if 


CENTRE  OF  GRAVITY. 


179 


the  wall  be  of  stone,  whose  specific  gravity  is  about  2520  ; 

then  -=§,  and  ./-=./  4 ==  -895  ; hence  fe  = -358  ae 
n n vo 

= TST  ae  : that  is,  when  the  rectangular  wall  is  of  stone,  the 
breadth  must  be  at  least  T\  of  its  height. 

236.  But  if  the  figure  of  the  wall  /, 

be  a triangle,  the  outer  side  tapering 
to  a point  at  top  Then  the  lever 
fn=|fe,  and  the  area  a = -*-fe  . ae  ; 
consequently  its  effort  a . n . fn  is  = 

Afe2  . ae  . n ; which  being  put  = 

AE2  . ab2  . 

— — m,  the  equation  gives  fe  = 


A-E  v/-  = A«iy-  for  the  breadth 
eb  2n  v 2n 

©f  the  wall  at  the  bottom,  for  an  equilibrium  in  this  case  also. 
— If  the  angle  of  the  slope  e be  45°  ; then  will  fe  be  = 

A E OYl  'lYb 

— , -J  — = 4ae./ — . And  when  this  wall  is  of  brick,  then 
sf  2 2 n v 


fe  = 4ae  nearly  But  when  it  is  of  stone  ; then  4-/-  = 

“ U 

•447  = 4 nearly  ; that  is,  the  triangular  stone  wall  must 
have  its  thickness  at  bottom  equal  to  | of  its  height  And 
in  like  manner,  for  other  figures  of  the  wall  and  also  for 
other  figures  of  the  earth. 


PROPOSITION  XLVI. 

237.  To  determine  the  Thickness  of  a Pier,  necessary  to  sup - 
port  a given  Arch. 

Let  abcd  be  half 
the  arch,  and  defg  the 
pier.  From  the  centre 
of  gravity  k of  the  half 
arch  draw  kl  perp.  oa  ; 
also  okr,  and  tkqp 
perp.  to  it  ; also  draw 
lq  and  GF  perp  to  tp, 
or  parallel  to  okr. 

Then  if  kl  represent 
the  weight  of  the  arch 
BCDA,  in  the  direction  of  gravity,  this  will  resolve  into  kq, 
the  force  acting  against  the  pier  perp  to  the  joint  sr.  and 
bq  the  part  of  the  force  parallel  to  the  same.  Now  kq  de- 
notes 


180 


STATICS. 


notes  the  only  force 
acting  perp.  on  the  arm 
gp,  of  the  crooked  lever 
fgp,  to  turn  the  pier 
about  the  point  g ; con- 
seq  K(i  X gp  will  de- 
note the  efficacious  force 
of  the  arch  to  overturn 
the  pier. 

Again,  the  weight  of 
the  pier  is  as  the  area 
df  X fg  ; therefore  df. 
fg  . |fg,  or  ^df  . fg2.  is  its  effect  on  the  lever  £fg,  to  pre 
vent  the  pier  from  being  overset  ; supposing  the  length  of 
the  pier,  from  point  to  point,  to  be  no  more  than  the  thick- 
ness of  the  arch. 

But  that  the  pier  and  the  arch  may  be  in  equilibrio,  these 
two  efforts  must  be  equal  Therefore  we  have  Idf  fg2  — 

^--kG;  ' A,  an  equation,  by  which  will  be  determined  the 

thickness  of  the  pier  fg  ; a denoting  the  area  of  the  half 
arch  bcda*. 

Example  1.  Suppose  the  arc  aem  to  be  a semicircle  ; and 
that  cn  or  oa  or  ob  = 45,  bc  — 7 feet,  af  = 20.  Hence  ad  = 
52.  d f •=  ce  = 72  Also  By  measurement  are  found  ok  = 
50-3,  kl  = 40-6,  do  — 29-7,  td  ===  30  87,  kq  — 24,  the 
area  bcda  = 750  = a ; and  putting  fg  — x the  breadth 
of  the  pier- 

Then  te  = td  + de  = 30  87  + and  kl  : lo  : : te  : 
ev  = 22  58+0-73z, 
then  ge  — ev  gv  = 49-42  — -73*, 
lastly  ok  : kl  . : gv  : gp  = 39-89  — 59a-. 

These  values  being  now  substituted  in  the  theorem  4df. 

fg2  = g‘ve  36*2  = 17665  — 261-5.X,  or  * 2 + 


* Note ■ As  it  is  commonly  a troublesome  thing  to  calculate  the 
place  of  she  centre  of  gravity  k of  the  half  arch  adcb,  it  n a bc 
easily,  and  sufficiently  near,  found  mechanically  in  the  manner  des- 
cribed in  art.  211,  thus  : Construct  that  space  adcb  accurately  by  a 
scale  to  the  given  dimensions,  on  a plate  of  any  uniform  flat  sub- 
stance, or  even  card  paper  ; then  cut  it  nicely  out  by  the  extreme 
lines,  and  balance  it  over  any  edge  or  the  sides  of  a table  in  two  posi- 
tions, and  the  intersection  of  the  two  places  will  give  the  situation  of 
the  poin;  k ; then  the  distances  or  lines  may  be  measured  by  the  scale, 
excep  t hose  depending  on  the  breadth  of  the  pier  FG,  viz.  the  lines 
as  mentioned  in  the  examples. 


CT  ED  E 


STRENGTH  AND  STRESS  OF  BEAMS,  &c.  181 


7-26x  = 490-7  ; the  root  of  which  quadratic  equation  gives 
x = 18-8  feet  = de  or  fg,  the  thickness  of  the  pier  sought. 

Example  2.  Suppose  the  span  to  be  109  feet,  the  height 
40  feet,  the  thickness  at  the  top  6 feet,  and  the  height  of  the 
pier  to  the  springer  20  feet,  as  before. 

Here  the  fig. 
may  be  considered 
as  a circular  seg- 
ment, having  the 
versed  sine  ob  = 

40,  and  the  right 
3ine  oa  or  oc  = 

50  ; also  bd  — 6, 
cf  = 20,  and  ef  = 

66.  Now,  by  the 
nature  of  the  cir- 


cle, whose  centre  is  w,  the  radius  wb  = 

,ob  • f-o'd2  40*  4-  50 2 r , , L _r,, 

— ^ J = 51i;  hence  ow  = 511  — 40=  111; 

and  the  area  of  the  semi-segment  obc  is  found  to  be  1491  ; 
which  is  taken  from  the  rectangle  odec  = od  . oc  = 46  x 50  = 
2300,  there  remains  809  = a,  the  area  of  the  space  bdecb. 
Hence,  by  the  method  of  balancing  this  space,  and  measur- 
ing the  lines,  there  will  be  found,  kc  = 18,  ik  — 34-6,  ix  = 
42,  kx  = 24,  ox  = 8,  iq  = 19-4,  te  = 35  6,  and  th  = 
35-6  + x,  putting  x — eh,  the  breadth  of  the  pier.  Then 
ik  : kx  : : th  : hv  — 24-7  -f-  0-7x  ; hence  gh  — hv  = 
41-3-0-7  = GV*  and  ix  : ik  : : gv  : gp  = 34-02  — 0.58.x. 
These  values  being  now  substituted  in  the  theorem  Ief. 

, gives  33x2  = 15431-47  — 253x,  or  x3  -f- 

Bx  -=  467-62,  the  root  of  which  quadratic  equation  gives 
x — 18  = eh  or  fg,  the  breadth  of  the  pier,  and  which  is 
probably  very  near  the  truth. 


FG2  = 


IQ.  GP  A 
IK 


ON  THE  STRENGTH  AND  STRESS  OF  BEAMS  OR 
BARS  OF  TIMBER  AND  METAL,  &c. 


238.  Another  use  of  the  centre  of  gravity,  which  may  be 
here  considered,  is  in  determining  the  strength  and  the 
stress  of  beams  and  bars  of  timber  and  metal,  &c.  in  differ- 
ent positions  ; that  is,  the  force  or  resistance  which  a beam 
or  bar  makes,  to  oppose  any  exertion  or  endeavour  made  to 
break  it  : «md  the  force  or  exertion  tending  to  b/eak  it  ; 

both 


182 


STATICS. 


both  of  which  will  be  different,  according  to  the  plaee  and 
position  of  the  centres  of  gravity. 

i PROPOSITION  XLVII. 

239.  The  Absolute  Strength  of  any  Bar  in  the  Direction  of  its 

Length , is  Directly  Proportional  to  the  Area  of  its  Trans - 

verse  Section. 

. 

Suppose  the  bar  to  be  suspended  by  one  end,  and  hanging 
freely  in  the  manner  of  a pendulum  ; and  suppose  it  to  be 
strained  in  direction  of  its  length,  by  any  force,  or  weight  j 
acting  at  the  lower  part,  in  the  direction  of  that  length,  suf- 
ficient to  break  the  bar,  or  to  separate  all  its  particles  Now, 
as  the  straining  force  acts  in  the  direction  of  the  length  all 
the  particles  in  the  transverse  section  of  the  body,  where  it 
breaks,  are  equally  strained  at  the  same  time;  and  they  must 
all  separate  or  break  together,  as  the  bar  is  supposed  to  be  of 
uniform  texture.  Thus  then,  the  particles  all  adhering  and 
resisting  with  equal  force,  the  united  strength  of  the  whole, 
will  be  proportional  to  the  number  of  them,  or  as  the  trans- 
verse section  at  the  fracture. 

240.  Corol.  1.  Hence  the  various  shapes  of  bars  make  no 
difference  in  their  absolute  strength  ; this  depending  only 
on  the  area  of  the  section,  and  must  be  the  same  in  all  equal 
areas,  whether  round,  or  square,  or  oblong,  or  solid,  or  hol- 
!ow,  &c. 

241.  Corol.  2.  Hence  also,  the  absolute  strengths  of  dif- 
ferent bars,  of  the  same  materials,  are  to  each  other  as  their 
transverse  sections,  whatever  their  shape  or  form  may  be. 

242.  Corol.  3.  The  bar  is  of  equal  strength  in  every  part 
of  it,  when  of  any  uniform  thickness,  or  prismatic  shape, 
and  is  equally  liable  to  be  drawn  asunder  at  any  part  of  its 
length,  whatever  that  length  may  be,  by  a weight  acting  at 
the  bottom,  independent  of  the  weight  of  the  bar  itself ; but 
when  considered  with  its  own  weight,  it  is  the  more  disposed 
to  break,  and  with  the  less  additional  appended  w'eight,  the 
longer  the  bar  is  on  account  of  its  own  weight  increasing 
with  its  length  And,  for  the  same  reason,  it  will  be  more 
and  more  liable  to  be  broken  at  every  point  of  its  length,  all 
the  way  in  ascending  or  counting  from  the  bottom  to  the 
top,  where  it  may  always  be  expected  to  part  asunder.  And 
hence  we  see  the  reason  w hy  longer  bars  are,  in  this  way 
more  liable  to  break  than  shorter  ones,  or  with  less  ap- 
pended weights.  Hence  also  we  perceive  that,  by  gradually 
increasing  these  weights,  till  the  bar  separates  and  breaks, 

then 


STRENGTH  AND  STRESS  OF  BEAMS,  &c.  383 


then  the  last  or  greatest  weight,  is  the  proper  measure  of  the 
absolute  strength  of  the  bar.  And  the  same  is  the  case  with 
a rope,  or  cord.  &c. — So  much  then  for  the  longitudinal 
strength  and  sfress  of  bodies.  Proceed  we  now  to  consider 
those  of  their  transverse  actions. 

PROPOSITION  XLYIIX 

243.  The  Strength  of  a Beam  or  Bar , of  Wood  or  Metal , fyc.  in. 
a Lateral  or  Tranverse  Direction , to  resist  a Force  acting  La- 
terally, is  Proportional  to  the  Area  or  Section  of  the  Bearn  in  that 
Place , Drawn  into  the  Distance  of  its  Centre  of  Gravity  from 
the  Place  where  the  Force  acts,  or  where  the  Fracture  will  end. 

Let  ab  represent  the  beam 
or  bar,  supported  at  its  two 
ends,  and  on  which  is  laid  a 
weight  w,  to  cause  a trans- 
verse fracture  abee.  The  force 
w acting  downwards  there,  the 
fracture  will  commence  or  open 
across  the  fibres,  in  the  oppo- 
site or  lowest  line  ab  ; from 
thence,  as  the  weight  presses  down  the  upper  line  ee,  the 
fracture  will  open  more  and  more  below,  and  extend  gradually 
upwards,  successively  to  the  parallel  lines  of  fibres  cc,  dd,  &c. 
till  it  arrive  at,  and  finally  open  in  the  last  line  of  fibres  ee, 
where  it  ends  ; when  the  whole  fracture  is  in  the  form  of  a 
wedge  widest  at  the  bottom,  and  ending  in  an  edge  or  line  ee 
at  top.  Now  the  area  ae  contains  and  denotes  the  sum  of  all  the 
fibres  to  be  broken  or  torn  asunder  ; and  as  they  are  suppos- 
ed to  be  all  equal  to  one  another,  in  absolute  strength,  that 
area  will  denote  the  aggregate  or  rvhole  strength  of  all  the  fi- 
bres in  the  longitudinal  direction,  as  in  the  foregoing  proposi- 
tion. But,  with  regard  to  lateral  strength,  each  fibre  must  be 
considered  as  acting  at  the  extremity  of  a lever  whose  centre 
of  motion  is  in  the  line  ee  : thus,  each  fibre  in  the  line  ab,  will* 
resist  the  fracture,  by  a force  proportional  to  the  product  of 
its  individual  strength  into  its  distance  ae  from  the  centre  of 
motion  consequently  the  resistance  of  all  the  fibres  in  ab,  will 
be  expressed  by  ab  X ae.  In  like  manner,  the  aggregate  re- 
sistance of  another  course  of  fibres,  parallel  to  ab,  as  cc,  will 
be  denoted  by  cc  X ce  ; and  a third,  as  dd,  by  ddy.de;  and 
so  on  throughout  the  whole  fracture  So  that  the  sum  of 
all  these  products  will  express  the  total  strength  or  resistance 

of 


184 


STATICS. 


of  all  the  fibres  or  of  the  beam  in  that  part.  But,  by  art, 
222,  the  sum  of  all  these  products  is  equal  to  the  product 
of  the  area  aeeb,  into  the  distance  of  its  centre  of  gravity  from 
ee.  Hence  the  proposition  is  manifest. 

244.  Corol.  1.  Hence  it  is  evident  that  the  lateral  strength 
of  a bar,  must  be  considerably  less  than  the  absolute  longi- 
tudinal strength  considered  in  the  former  proposition,  and  will 
be  broken  by  a much  less  force,  than  was  there  necessary  to 
draw  the  bar  asunder  lengthways.  Because,  in  the  one  case 
the  fibres  must  be  all  separated  at  once,  in  an  instant ; but  in 
the  other,  they  are  overcome  and  broken  successively,  one 
after  another,  and  in  some  portion  of  time.  For  instance, 
take  a walking  stick,  and  stretching  it  lengthways,  it  will  bear  a 
very  great  force  before  it  can  be  drawn  asunder  ; but  again 
taking  such  a stick,  apply  the  middle  of  it  to  the  bended  knee, 
and  with  the  two  hands  drawing  the  end  towards  you,  the  stick 
is  broken  across  by  a small  force. 

245  Corol.  2.  In  square  beams,  the  lateral  strengths  are 
as  the  cubes  of  the  breadths  or  depths. 

246.  Corol.  3.  And  in  general,  the  lateral  strengths  of  any 
bars,  whose  sections  are  similar  figures,  are  as  the  cubes  of 
the  similar  sides  of  the  sections. 

247.  Corol.  4.  In  cylindrical  beams,  the  lateral  strengths 
are  as  the  cubes  of  the  diameters. 

248.  Corol.  5.  In  rectangular  beams,  the  lateral  strengths 
are  to  each  other,  as  the  breadths  and  square  of  the  depths. 

249.  Corol.  6.  Therefore  a joist  laid  on  its  narrow  edge,  is 
stronger  than  when  laid  on  its  flat  side  horizontal,  in  propor- 
tion as  the  breadth  exceeds  the  thickness.  Thus  if  a joist  be 
10  inches  broad,  by  2|  thick,  then  it  will  bear  4 times  more 
when  laid  on  edge,  than  when  laid  flat.  Which  shows  the  pro- 
priety of  the  modern  method  of  flooring  with  very  thin,  but 
deep  joists. 

250.  Corol.  7.  If  a b$am  be  fixed  firmly  by  one  end  into  a 
wall,  in  a horizontal  position,  and  the  fracture  be  caused  by  a 
weight  suspended  at  the  other  end,  the  process  would  be  the 
same,  only  that  the  fracture  would  commence  above,  and  ter- 
minate at  the  lower  side  ; and  the  prop,  and  all  the  corollaries 
would  still  hold  good. 

251.  Corol.  8.  When  a cylinder  or  prism  is  made  hollow,  it 
is  stronger  than  when  solid,  with  an  equal  quantity  of  mate- 
rials 


STRENGTH  AND  STRESS  OF  BEAMS,  fee.  185 


rials  and  length,  in  the  same  proportion  as  its  outer  diameter 
is  greater.  Which  shows  the  wisdom  of  Providence  in  mak- 
ing the  stalks  of  corn,  and  the  feathers  and  bones  of  animals, 
&c.  to  be  hollow.  Also,  if  the  hollow  beam  have  the  hollow 
or  pipe  not  in  the  middle,  but  nearest  to  that  side  where  the 
fracture  is  to  end,  it  will  be  so  much  the  stronger. 

252.  Corol.  9.  if  the  beam  be  a triangular  prism,  it  will  be 
strongest  when  laid  with  the  edge  upwards,  if  the  fracture 
commence  or  open  first  on  the  under  side  ; otherwise  with 
the  flat  side  upwards  ; because  in  either  case  the  centre  of 
gravity  is  the  farther  from  the  ending  of  the  fracture  And 
the  same  thing  is  true,  and  for  the  same  reason,  for  any  other 
shape  of  the  prism  On  the  same  account  also,  a square  beam 
is  stronger  when  laid,  or  when  acting  angle-wise,  than  when 
on  a flat  side. 

PROPOSITION  XUX. 

253.  The  Lateral  Strengths  of  Prismatic  Beams,  of  the  same 
materials,  are  Directly  as  the  Areas  of  the  Sections  and,  the 
Distances  of  their  Centres  of  Gravity  ; and  Inversely  as 
their  Lengths  and  Weights. 

Let  ab  and  cn  represent  the 
two  beams  fixed  horizontally, 
by  their  ends,  into  an  upright 
wall  ac  Now,  by  the  last 
prop  the  strength  of  either 
beam,  considered  as  without  or 
independent  of  weight  is  as  its  section  drawn  into  the  distance 
of  its  centre  of  gravity  from  the  fixed  point,  viz.  as  sc,  where 
s denotes  the  transverse  section  at  a or  c,  and  c the  distance 
of  its  centre  of  gravity  above  the  lowest  point  a or  c.  But 
the  effort  of  their  weight,  w ot<w,  tending  to  separate  the  fi- 
bres and  break  the  beam,  are,  by  the  principle  of  the  lever, 
as  the  weight  drawn  into  the  distance  of  the  place  where  it 
may  be  supposed  to  be  collected  and  applied,  which  is  in  the 
middle  of  the  length  of  the  beam  ; that  is,  the  effort  of  the 
weight  upon  the  beam  is  as  wX|ab.  Hence  the  prop,  is 
manifest. 

'254.  Corol.  1.  Any  extraneous  weight  or  force  also,  any- 
where applied  to  the  beam,  will  have  a similar  effect  to  break 
the  beam  as  its  own  weight ; that  is,  its  effect  will  be  as  w X 
d,  as  the  weight  drawn  into  the  length  of  lever  or  distance 
from  a where  it  is  applied. 


Vol.  II. 


255.  Corol. 


186 


STATICS. 


255.  Carol.  2.  When  the  beam  is  fixed  at  both  ends,  the 
same  property  will  hold  good,  with  this  difference  only,  that 
in  this  case  the  beam  is  of  the  same  strength,  as  another  of  an 
equal  section,  and  only  half  the  length,  when  fixed  only  at  one 
end  For,  if  the  longer  beam  were  bisected,  or  cut  in  halves, 
each  half  wmuld  be  in  the  same  circumstances  with  respect  to 
its  fixed  end,  as  the  shorter  beam  of  equal  length. 

256.  Corol.  3.  Square  prisms  and  cylinders  have  their  lat- 
eral strengths  proportional  to  the  cubes  of  the  depths,  or  di- 
ameters, directly,  and  to  their  lengths  and  weights  inversely. 

Corol.  4.  Similar  prisms  and  cylinders  have  their  strengths 
inversely  proportional  to  their  like  linear  dimensions,  the 
smaller  being  comparatively  larger  in  that  proportion.  For 
their  strength  increases  as  the  cube  of  the  diameter  or  of  their 
length  ; but  their  stress,  from  their  weight  and  length  of  le- 
ver, as  the  4th  power  of  the  length. 

257.  Scholium.  From  the  foregoing  deductions  it  follows 
that,  in  similar  bodies  of  the  same  texture,  the  force  which 
tends  to  break  them,  or  to  make  them  liable  to  injury  by  ac- 
cidents, in  the  larger  bodies,  increases  in  a higher  proportion 
than  the  force  which  tends  to  preserve  them  entire,  or  to  se- 
cure them  against  such  accidents  ; their  disadvantage,  or  ten- 
dency to  break  by  their  own  weight,  increasing  in  the  same 
proportion  as  their  length  increases  : so  that,  though  a smaller 
beam  may  be  firm  and  secure,  yet  a large  and  similar  one  may 
be  so  long  as  to  break  by  its  own  weight.  Hence,  it  is  justly 
concluded,  that  what  may  appear  very  firm  and  successful  in 
a model  or  small  machine,  may 'be  weak  and  infirm,  or  even 
fall  in  pieces  by  its  own  weight,  when  it  is  executed  on  large 
dimensions  according  to  the  model. 

For,  in  similar  bodies,  or  engines,  or  in  animals,  the  greater 
must  be  always  more  liable  to  accidents  than  the  smaller,  and 
have  a less  relative  strength,  that  is,  the  greater  have  not  a 
strength  in  so'great  a proportion  as  their  magnitude.  A great- 
er column,  for  instance,  is  in  much  more  danger  of  breaking 
by  a fall,  than  a similar  smaller  one.  A man  is  in  more  danger 
from  accidents  of  this  kind  than  a child.  An  insect  can  bear 
and  carry  a load  many  times  heavier  than  itself ; whereas  a 
larger  animal,  as  a horse,  for  instance,  can  hardly  support  a 
a burden  equal  to  his  own  weight. 

From  the  same  principle  it  is  also  justly  inferred,  that 
there  are  necessarily  limits  in  all  the  works  of  nature  and 

art, 


STRENGTH  AND  STRESS  OF  BEAMS,  &c. 


187 


art,,  which  they  cannot  surpass  in  magnitude.  Thus,  for  in- 
stance, were  trees  to  be  of  a very  enormous  size,  their  branch- 
es would  break  and  fail  off  by  their  own  weight.  Large  an- 
imals have  not  strength  in  proportion  to  their  size  : and  if 
there  were  any  land  animals  much  larger  than  those  we  know, 
they  would  hardly  be  able  to  move,  and  would  be  perpetually 
subjected  to  most  dangerous  accidents. 

As  to  the  sea  animals  indeed,  the  case  is  different,  as  the 
pressure  of  the  water  in  a great  measure  sustains  them  ; and 
accordingly  we  find  they  are  vastly  larger  than  land  animals. 

From  what  has  been  said  it  clearly  follows  that  to  make 
bodies  or  engines,  or  animals,  of  equal  relative  strength,  the 
larger  ones  must  have  grosser  proportions,  or  a higher  de- 
gree of  thickness,  than  they  have  of  length.  And  this  senti- 
ment being  suggested  to  us  by  continual  experience,  we  natu- 
rally join  the  idea  of  greater  strength  and  force  with  the  gross- 
er proportions,  and  of  agility  with  the  more  delicate  ones.  In 
architecture,  where  the  appearance  of  solidity  is  no  less  re- 
garded than  real  firmness  and  strength,  in  order  to  satisfy  a 
judicious  eye  and  taste,  the  various  orders  of  the  columns 
serve  to  suggest  different  ideas  of  strength.  But,  by  the 
same  principle,  if  we  should  suppose  animals  vastly  large, 
from  the  gross  proportions  a heaviness  and  unwieldiness 
would  arise,  which  would  make  them  useless  to  themselves, 
and  disagreeable  to  the  eye.  In  this,  as  in  all  other  cases, 
whatever  generally  pleases  taste,  not  vitiated  by  prejudice 
of  education,  or  by  fabulous  and  marvellous  relations,  may 
be  traced  till  it  appears  to  have  a just  foundation  in  nature. 

PROPOSITION  L. 

258.  If  a Weight  be  placed , or  a Force  act,  on  any  part  of  a 
Horizontal  beam,  supported  at  both  e?ids,  the  Stress  upon  that 
part  will  be  as  the  Rectangle  or  Product  of  its  two  Distances 
from  the  supported  ends. 

That  is,  the  stress  upon  the 
beam  ab,  at  c,  by  the  weight  w, 
is  as  ac  X bc.  For,  by  the  na- 
ture of  the  lever,  the  effect  of 
the  weight  w,  on  the  lever  ac, 
is  ac  w ; and  the  effect  of  this 
force  acting  at  c,  on  the'  lever 

BC,  is  AC  . W . BC  = AC  . BC  . W. 

And,  the  weight  w being  given,  the  effect  or  stress  is  as  ac  . 

3C.  v 

259.  Corol. 


) 


188 


STATICS. 


259.  Cor ol.  1.  The  greatest  stress  is  when  the  weight  w 
is  at  the  middle  : for  then  the  rectangle  of  the  two  halves, 
ac  . ac  = iAB  |ab  = jab2  , is  the  greatest.  And  from  the 
middle  point,  the  3tress  is  less  and  less  all  the  way  to  the  ex- 
tremities a and  b,  where  it  is  nothing. 

260.  Corol.  2.  The  same  thing  will  obtain  from  the  weight 
of  the  beam  itself,  or  from  any  other  weight  diffused  equally 
all  over  it  ; the  stress  in  this  case  being  the  half  of  the 
former.  So  that,  in  all  structures,  we  should  avoid  as  much 
as  possible,  placing  weights  or  strains  in  the  middle  of 
beams. 

261.  Corol  3.  If  ® be  the  greatest  weight  that  a beam  can 
sustain  at  its  middle  point  ; and  it  be  required  to  find  the 
place  where  it  will  support  any  greater  weight  w ; that  point 
will  be  found  by  making,  as  w : w : : Aab  . Lab,  or  £ab2  ; 
AC  . EC  or  AC  X (aB  — ac)  = AB  . AC  — AC2. 

PROPOSITION  LI. 

262.  When  a Beam  is  placed  aslope,  its  Strength  in  that  position, 
is  to  its  Strength  when  Horizontal,  to  resist  a Vertical  Force, 
as  the  Square  of  Radius  is  to  the  Square  of  the  Cosine  of  the 
Elevation. 

Let  ab  be  the  beam  standing  aslope, 
cf  prep,  to  the  horizon  afg  ; then  cd 
is  the  vertical  section  of  the  beam,  and 
ce,  prep,  to  ab,  is  the  transverse  sec- 
tion, and  is  the  same  as  when  in  the 
horizontal  position  Now,  the  strength, 
in  both  positions,  is  as  the  section  drawn 
into  the  distance  of  its  centre  of  gravity 
from  the  point  c.  But  the  sections,  be-  F G 

ing  of  the  same  breadth,  are  as  their 
depths,  cd,  ce  ; and  the  distances  of  the  centres  of  gravity 
are  as  the  same  depths  ; therefore  the  strengths  are  as  cd  . 
cd  to  ce  . ce,  or  cd2  to  ce2.  But,  by  the  similar  triangles 
cde,  afd,  it  is  cd  : ce  : : ad  : af,  as  radius  to  the  cosine  of 
the  elevation.  Therefore  the  oblique  strength  is  to  the  trans- 
verse strength,  as  ad2  to  af2,  the  square  of  radius  to  the 
square  of  the  cosine  of  elevation. 

263.  Corol.  1.  The  strength  of  a beam  increases  from  the 
horizontal  position,  where  it  is  least,  all  the  way  as  it  revolves 
to  the  vertical  position,  where  it  is  the  greatest. 


PRO. 


STRENGTH  AND  STRESS  OF  BEAMS,  &c.  189 


PROPOSITION  LH. 

264.  When  Beams  stand  Aslope,  or  Obliquely,  and  sustaining 
Weights,  either  at  the  Middle  Points,  or  in  any  other  Similar 
Situations,  or  Equally  Diffused  over  their  Lengths  ; the  Strains 
upon  them  are  Directly  as  the  Weights,  and  the  Lengths,  and 
the  Cosines  of  Elevation. 

For,  by  the  inclined  plane,  the  weight  is  to  the  pressure 
on  the  plane,  as  ac  to  af,  as  radius  to  the  cosine  of  elevation  : 
therefore  the  pressure  is  as  the  weight  drawn  into  the  cosine 
of  the  elevation.  Hence  the  stress  will  be  as  the  length  of 
the  beam  and  this  force  ; that  is,  as  the  weight  X length  X 
cosine  of  elevation. 

265.  Corol.  1.  When  the  lengths  and  weights  of  beams 
are  the  same,  the  stress  is  as  the  cosine  of  elevation  ; and  it 
is  therefore  the  greatest  when  it  lies  horizontal. 

266.  Corol.  2.  In  all  similar  positions,  and  the  weights  vary- 
ing as  the  lengths,  or,  the  beams  uniform  ; then  the  stress  varies 
as  the  squares  of  the  lengths. 

267.  Corol.  3.  When  the  weights 
are  equal,  on  the  oblique  beam  ab, 
and  the  horizontal  one  ac,  and  bc 
is  vertical  : the  stress  on  both  beams 
is  equal.  For,  the  length  into  the 
cosine  of  elevation  is  the  same  in 
both  ; or  ab  X cos.  a=ac  X ra- 
dius. 

268.  Corol.  4.  But  if  the  weights  on  the  beams  vary  as 
their  lengths  ; then  the  stress  will  also  vary  in  the  same 
ratio. 

269.  Corol.  5.  And  universally,  the  stress  upon  any  point 
of  an  oblique  beam,  is  as  the  rectangle  of  the  segments  of  the 
beam,  and  the  weight,  and  cosine  of  inclination,  directly  ; and 
the  length  inversely. 


PRO. 


190 


STATICS. 


PROPOSITION  LIU. 

270.  When  a Beam  is  to  sustain  any  Weight,  or  Pressure,  or 
Force,  acting  Laterally  ; then  the  Strength  ought  to  be  as 
the  Stress  upon  it  ; that  is,  the  Breadth  multiplied  by  the 
Square  of  the  Depth,  or  in  similar  sections,  the  Cube  of  the 
Diameter,  in  every  place,  ought  to  be  proportional  to  the 
Length  drawn  into  the  Weight  or  Force  acting  on  it.  And  the 
same  is  true  of  several  Different  Pieces  of  timber  compared 
together. 

For  every  several  piece  of  timber  or  metal,  as  well  as 
every  part  of  the  same,  ought  to  have  its  strength  propor- 
tioned to  the  weight,  force,  or  pressure  it  is  to  support.  And 
therefore  the  strength  ought  to  be  universally,  or  in  every 
part  as  the  stress  upon  it.  But  the  strength  is  as  the  breadth 
into  the  square  of  the  depth  ; and  the  stress  is  as  the  weight 
or  force  into  the  distance  it  acts  at.  Therefore  these  must  be 
in  constant  ratio.  This  general  property  will  give  rise  to  the 
effect  of  different  shapes  in  beams,  according  to  particular  cir- 
cumstances ; as  in  the  following  corollaries. 

271.  Corol.  1.  If  abc  be  a hori- 
zontal beam,  fixed  at  the  end  ac, 
and  sustaining  a weight  at  the  other 
end  b.  And  if  the  sections  at  all 
places  be  similar  figures  ; and  de  be 
the  diameter  at  any  place  d ; then 
bd  will  be  every  where  as  de3.  So  that  if  adb  be  a right  line, 
then  bec  will  be  a cubic  parabola.  In  which  case§  of  such  a 
beam  may  be  cut  away,  without  any  diminution  of  the 
strength. — But  if  the  beam  he  bounded  by  two  parallel  planes, 
perpendicular  to  the  horizon  ; then  bd  will  be  as  de2  ; and 
then  bec  will  be  the  common  parabola  in  which  case  a 3d  part 
of  the  beam  may  be  thus  cut  away. 

272.  Corol.  2.  But  if  a weight  press  uniformly  on  every  part 
of  ab  ; and  the  sections  in  all  points,  as  d,  be  similar  ; then  bd2 
will  be  every  where  as  de3  : and  then  bec  is  the  semicubical 
parabola. 

But.  in  this  disposition  of  the 
weight,  if  the  beam  be  bounded  by 
parallel  planes, perpendicular  to  the 
horizon  ; then  bd  will  be  always  as 
de  ; and  bec  a right  line,  or  abc  a 
wedge.  So  that  then  half  the  beam 
may  be  cutaway,  without  diminution  of  strength. 

273.  Corol. 


STRENGTH  AND  STRESS  OF  BEAMS,  &c.  191 


273.  Corol.  3.  -If  the  beam  ab 
be  supported  at  both  ends  ; and 
if  it  sustain  a weight  at  any  va- 
riable point  d,  or  uniformly  on 
all  parts  of  its  length  ; and  if  all  the  sections  be  similar 
figures  ; then  will  the  diameter  de3  be  every  where  as  the 
rectangle  ad  . db, 

But  if  it  be  bounded  by  two  parallel  planes,  perpendicular 
to  the  horizon  ; then  will  de2  be  every  where  as  the  rect- 
angle ad  . db,  and  the  curve  aeb  an  ellipsis. 


274.  Corol.  4.  But  if  a weight 
be  placed  at  any  given  point  f, 

and  all  the  sections  be  similar  fig-  A D F B 

ures  ; then  will  ad  be  as  de3, 

and  ac,  bg  be  two  cubic  parabo-  E G- 

las. 

But  if  the  beam  be  bounded  by  two  parallel  planes,  per- 
pendicular to  the  horizon  ; then  ad  is  as  de2,  and  ag  and 
bg  are  two  common  parabolas. 


275.  Scholium.  The  relative  strengths  of  several  sorts  of 
wood,  and  of  other  bodies,  as  determined  by  Mr.  Emerson, 
arexas  follow  : 


Iron 107 

Brass 50 

Bone  -------  22 

Box,  Yew,  Plumbtree,  Oak  - if 

Elm,  Ash  ------- 

Walnut,  Thorn  ------  7I 

Red  fir,  Holly,  Elder,  Plane,  Crabtree,  Appletree  7~ 

Beech,  Cherrytree,  Hazle  - 62- 

Lead  -------  gr 

Alder,  Asp,  Birch,  White  fir,  Willow’  - (f 

Fine  freestone  ------  j 

A cylindric  rod  of  good  clean  fir,  of  1 inch  circumference, 
drawn  lengthways,  will  bear  at  extremity  400  lbs  ; and  a 
spear  of  fir,  2 inches  diameter,  will  bear  about  7 tons  in  that 
direction. 

A rod  of  good  iron,  of  an  inch  circumference,  will  bear 
a stretch  of  near  3 tons  weight. 

A good  hempen  rope,  of  an  inch  circumference,  will  bear 
1000  lbs  at  the  most. 

Plence  Mr.  Emerson  concludes,  that  if  a rod  of  fir,  or  of 

- iron. 


192 


STATICS.  ' 


iron,  or  a rope  of  d inches  diameter,  were  to  lift  a of  the  ex- 
treme weight  ; then 

The  fir  would  bear  8|  d2  hundred  weights. 

The  rope  - - 22  d2  ditto. 

The  iron  - - d2  tons. 

Mr.  Banks,  an  ingenious  lecturer  on  mechanics,  made 
many  experiments  on  the  strength  of  wood  and  metal  ; 
whence  he  concludes,  that  cast  iron  is  from  3i  to  \\  times 
stronger  than  oak  of  equal  dimensions  ; and  from  5 to 
times  stronger  than  deal.  And  that  bars  of  cast  iron,  an 
inch  square,  weighing  9 lbs.  to  the  yard  in  length,  supported 
at  the  extremities,  bear  on  an  average,  a load  of  970  lbs. 
laterally.  And  they  bend  about  an  inch  before  they  break. 

Many  other  experiments  on  the  strength  of  different  ma- 
terials, and  curious  results  deduced  from  them,  may  be  seen 
in  Dr.,  Gregory’s  and  Mr  Emerson’s  Treatises  on  Mechanics, 
as  well  as  some  more  propositions  on  the  strength  and  stress 
of  different  bars. 


ON  THE  CENTRES  OF  PERCUSSION,  OSCILLATION, 
AND  GYRATION. 

276.  THE  Centre  of  Percussion  of  a body,  or  a system 
of  bodies,  revolving  about  a point,  or  axis,  is  that  point,  which 
striking  an  immoveable  object,  the  whole  mass  shall  not  incline 
to  either  side,  but  rest  as  it  were  in  equilibrio,  without  acting 
on  the  centre  of  suspension. 

277.  The  Centre  of  Oscillation  i9  that  point,  in  a body 
vibrating  by  its  gravity,  in  which  if  any  body  be  placed,  or  if 
the  whole  mass  be  collected,  it  will  perform  its  vibrations  in 
the  same  time,  and  with  the  same  angular  velocity,  as  the 
whole  body,  about  the  same  point  or  axis  of  suspension 

278.  The  Centre  of  Gyration,  is  that  point,  in  which  if 
the  whole  mass  be  collected,  the  same  angular  velocity  will 
be  generated  in  the  same  time,  by  a given  force  acting  at  any 
place,  as  in  the  body  or  system  itself. 

279.  The  angular  motion  of  a body-,  or  system  of  bodies, 
is  the  motion  of  a line  connecting  any  point  and  the  centre 
or  axis  of  motion  ; and  is  the  same  in  all  parts  of  the  same 
revolving  body.  And  in  different  unconnected  bodies,  each 
revolving  about  a centre,  the  angular  velocity  is  as  the  abso- 
lute velocity  directly,  and  as  the  distance  from  the  centre 
inversely  ; so  that,  if  their  absolute  velocities  be  as  their 
radii  or  distances,  the  angular  velocities  will  be  equal. 


PROP- 


CENTRE  OF  PERCUSSION. 


193' 


PROPOSITION  LTV. 


!80.  To  find  the  Centre  of  Percussion  of  a Body , or  System 
of  Bodies. 


Let  the  body  revolve  about  an  axis 
passing  through  any  point  s in  the  line 
sgo,  passing  through  the  centres  of  gra- 
vity and  percussion,  g and  o.  Lee  mk 
be  the  section  of  the  body,  or  the  plane 
in  which  the  axis  sgo  moves.  And 
eonceive  all  the  particles  of  the  body  to 
be  reduced  to  this  plane,  by  perpendi- 
culars let  fall  from  them  to  the  plane  : a 
supposition  which  will  not  affect  the 
centres  g,  o,  nor  the  angular  motion  of 
the  body. 


Let  a be  the  place  of  one  of  the  particles,  so  reduced  ; 
join  sa,  and  draw  ap  perpendicular  to  as,  and  Aa  perpendi- 
cular to  sgo  : then  ap  will  be  the  direction  of  a’s  motion  as 
it  revolves  about  s ; and  the  whole  mass  being  stopped  at  o, 
the  body  a will  urge  the  point  p,  forward,  with  a force  pro- 
portional to  its  quantity  of  matter  and  velocity,  or  to  its 
matter  and  distance  from  the  point  of  suspension  s ; that  is, 
as  a . sa  ; and  the  efficacy  of  this  force  in  a direction  per- 
pendicular to  so,  at  the  point  p,  is  as  a . sa,  by  similar  tri- 
angles ; also,  the  effect  of  this  force  on  the  lever,  to  turn  it 
about  o,  being  as  the  length  of  the  lever,  is  as  a . sa  . po  = 
i . sa  . (so  — sp)  ==  a . sa  . so  — a . sa  . sp  = a . sa  . so  — 
4 . sa2.  In  like  manner,  the  forces  of  b and  c,  to  turn  the 
system  about  o,  are  as 

b . sb  . so — b . sb2,  and 
c . sc  . so — c . sc2,  &c. 

But,  since  the  forces  on  the  contrary  sides  of  o destroy 
one  another,  by  the  definition  of  this  force,  the  sum  of  the 
positive  parts  of  these  quantities  must  be  equal  to  the  sum  of 
:he  negative  parts, 

sb  . so-f-c  . sc  . so  Stc.  = 
sb2  + c . sc2  &c-  ; and 


hat  is,  a 


sa 

A 


so  + B 
SA 


2 -j-  B 
A 


hence  so 
Vot.  II. 


SA2  -j-  B . SB2  -j-  C . SC2  &C. 


sa  + b . 
2S 


sb  + c . sc  &c. 


which 


194 


STATICS. 


which  is  the  distance  of  the  centre  of  percussion  below  the 
axis  of  motion. 

And  here  it  may  be  observed  that,  if  any  of  the  points 
a,  b,  &c  fall  on  the  contrary  side  of  s,  the  corresponding 
product  a . sa,  or  b . sb,  &c.  must  be  made  negative. 

281.  Corol.  1.  Since,  by  cor.  3,  pr.  40,  a + B + c &c. 
or  the  body  b X the  distance  of  the  centre  of  gravity,  sg, 
is  = a . sa  + b . sb  + c . sc  & c.  which  is  the  denominator 
of  the  value  of  so  ; therefore  the  distance  of  the  centre  of 

A SA2  -f  B . SB*  4 c ■ sc*  &c. 

percussion,  is  so  — . 

1 sg  X body  b 

282.  Corol.  2.  Since,  by  Geometry,  theor.  36,  37, 

it  is  sa2l=  sg2  + ga2  — 2sg  . Ga, 

and  sb-I=  sg2  + gb*  4*  2sg  . cb, 

and  sc2 1=  sg2  + gc2  4~  2sg  . gc,  &c  ; 

and,  by  cor.  5,!pr.  40,  the  sum  of  the  last  terms  is  nothing, 
namely,  — 2so" . ca  -f  2sg  . cb  -4  2sg  . gc  &c  = 0; 

therefore  the  sum  of  the  others,  or  a . sa2  + b . sb2  &c.  - 

is  = (a  4*  B &C.)  . SG2  4"  A . GA2  4'  B . GB2  4*  c • GC2  &C. 

or  = b . SG2  4"  A . GA2  -f  B . GB2  4 C . GC2  &c. ; 

which  being  substituted  in  the  numerator  of  the  foregoing 
value  of  so,  gives 

b . SG  4-  A . GA2  4 B • OB* 


so 


or 


SO  = SG  4* 


&C. 

b . SG  ’ 

A . GA2  4-  B . GB*  4-  C . GC*  &C. 

b . sg 


283  Corol.  3.  Hence  the  distance  of  the  centre  of  per- 
cussion always  exceeds  the  distance  of  the  centre  of  gravity. 

, , . , a . ga*  4 B . GB*  &c. 

and  the  excess  is  always  go  = — 


284.  And  hence,  also,'SG  . go  = 


b . SG 


A • GA!  4 b . gb*  Sec. 


the  body  b 

that  is  sg,  co  is  always  the  same  constant  quantity,  where-  i 
ever  the  point  of  suspension  s is  placed  ; since  the  point  g j 
and  the  bodies  a,  b,  &c.  are  constant.  Or  go  is  always 
reciprocally  as  sg.  that  is  go  is  less,  as  sg  is  greater  ; and 
consequently  the  point  o rises  upwards  and  approaches  to- 
wards the  point  g,  as  the  point  s is  removed  to  the  greater 
distance  ; and  they  coincide  when  sg  is  infinite.  But  when 
s coincides  with  g,  then  go  is  infinite,  or  o is  at  an  infinite 
distance. 

PROPOSITION 


CENTRE  OF  PERCUSSION. 


196 


PROPOSITION  LV. 


285.  If  a Body  a,  at  the  Distance  sa  from  an  axis  passing 
through  s,  be  made  to  revolve  about  that  axis  by  any  Force 
acting  at  p in  the  Line  sp,  Perpendicular  to  the  Axis  of  Motion: 
It  is  required  to  determine  the  Quantity  or  Matter  of  another 
Body  q,  which  being  placed  at  p,  the  Point  where  the  Force  acts, 
it  shall  be  accelerated  in  the  Same  Manner , as  when  a revolved 
at  the  Distance  sa;  and  consequently , that  the  Angular  Velocity 
of  a and  q about  s,  may  be  the  Same  in  Both  Cases. 

By  the  nature  of  the  lever,  sa  : sp  : :f  : 

SP 

— .f  the  effect  of  the  force  /,  acting  at  p, 

on  the  body  at  a ; that  is,  the  force  f acting  at 
p,  will  have  the  same  effect  on  the  body  a,  as 

the  force  — /,  acting  directly  at  the  point  a, 

S A 

But  as  asp  revolves  altogether  about  the  axis  at  s,  the  abso- 
lute velocities  of  the  points  a and  s,  or  of  the  bodies  a and 
0..  will  be  as  the  radii  sa,  sp,  of  the  circle  described  by  them. 
Here  then  we  have  two  bodies  a and  q which  being  urged 

directly  by  the  forces/ and  f acquire  velocities  which  are 

as  sp  and  sa.  And  since  the  motive  forces  of  bodies  are  as 
their  mass  and  velocity  : therefore  .... 

SP  SA2 

— f : f : : a . sa  : q . sp,  and  sp2  : sa2  : : a : q.  = a, 

SAJJ  SP2  ’ 

which  therefore  expresses  the  mass  of  matter  which,  being 
placed  at  p,  would  receive  the  same  angular  motion  from  the 
action  of  any  force  at  p,  as  the  body  a receives.  So  that  the 
resistance  of  any  body  a,  to  a force  acting  at  any  point  p,  is 
directly  as  the  square  of  its  distance  sa  from  the  axis  of  mo- 
tion, and  reciprocally  as  the  square  of  the  distance  sp  of  the 
point  where  the  force  acts. 

286.  Corol.  1.  Hence  the  force  which  accelerates  the  point 

• r-  • f • S P2 

p,  is  to  the  force  of  gravity,  as  — to  1,  or  as  f . sp3 

A . SA2  J 

to  A . SA2. 


287.  Corol.  2.  If  any  number  of  bodies 
a,  b,  c,  be  put  in  motion,  about  a fixed 
axis  passing  through  s,  by  a force  act- 
ing at  r ; the  point  p will  be  accele- 
rated in  the  same  manner,  and  conse- 
quently the  whole  system  will  have  the 
same  angular  velocity,  if  instead  of  the 


bodies 


m 


STATICS. 


bodies  a,  b,  c,  placed  at  the  distances  sa,  sb,  se,  there  be 

Sa2'  SB2  SC2 

substituted  the  bodies a, b, c:  these  being  col- 

lected  into  the  point  p.  And  hence,  the  moving  force  be- 
ing/,  and  the  matter  moved  being — — ; 

theref. - • is  the  accelerating  force  ; 

A SA-  “h  b ■ s b2  -f  c . sc2 

which  therefore  is  to  the  accelerating  force  of  gravity,  as 

f . SP2  to  A . SA2  + B . SB2  + c . SC2. 


288.  Carol.  3.  The  angular  velocity  of  the  whole  system 
of  bodies,  is  as . For  the  absolute 

A . SA2  -f  B . SB2  -t-  C • SC2 

velocity  of  the  point  p,  is  as  the  accelerating  force,  or  di- 
rectly as  the  motive  force  /,  and  inversely  as  the  mass 


A ■ SA2  8tc 
S P 2 


but  the  angular  velocity  is  as  the  absolute  velo- 


city directly,  and  the  radius  sp  inversely  ; therefore  the  an- 
gular velocity  of  p,  or  of  the  whole  system,  which  is  the  same 
f SB 

thing,  is  as irzr~ „ • 

® a . sa2  -j-  b • sb2  -f-  c • SC2 


PROPOSITION  LVI. 

289.  To  determine  the  Centre  of  Oscillation  of  any  Compound 

Mass  or  Body  mn,  or  of  any  System  of  Bodies  a,  b,  c,  4‘C. 

Let  mn  be  the  plane  of  vibration,  to  which  let  all  the 
matter  be  reduced,  by  letting  fall  perpendiculars  from  every 
particle,  to  this  plane  Let 
g be  the  centre  of  gravity, 
and  o the  centre  of  oscilla- 
tion ; through  the  axis  s 
draw  sgo,  and  the  horizon- 
tal line  s q ; then  from  every 
particle  a,  b,  c,  &c.  let  fall 
perpendiculars  aix,  a/),b6,  b q, 
cc,cr,to  these  two  lines;  and 
join  sa,  sb,  sc  ; also,  draw 
cm,  on,  perpendicular  to  iq. 

Now  the  forces  of  the  weights 
a,  b,  c,  to  turn  the  body 
about  the  axis,  are  a , sp,  e. 
sq,  — c . s r ; therefore,  by 
eor.  3,  prop.  55,  the  angular 


model. 


CENTRE  OF  OSCILLATION 


197 


motion  generated  by  all  these  forces  is 


A . Sp  -f-  B • s<7  — C sr 
A.  sA*  + B-  B2  -f  C .sc2 


Also;  the  angular  veloc.  any  particle  p,  placed  in  o,  generates 


• , , . ...  p . sn  sn,  sm 

m the  system,  by  its  weight,  is  — or  — or , be' 

J J ® ’ p . SO 2 SO2  so . so 

cause  of  the  similar  triangles  sg in,  son.  But,  by  the  prob- 
lem, the  vibrations  are  performed  alike  in  both  cases,  and 
therefore,  these  two  expressions  must  be  equal  to  each  other, 

that  is, = - ; and  hence 

SG  • SO  a • sa-  1-  B . SB  -f-  c . sc2 
sm  A • SA2  H-  B . -B  ' -f-  c . tC2 

SO  = X . 

SG  A sp  -p  B • s?  — c • sr 

But,  by  cor.  2,  pr  41,  the  sura  a . sp  --j-  b . sq  — c . sr  = 
(a  + b -j-  c)  . sm  ; therefore  the  distance  so  — - - - - 

A . s A 2 + B . 'B2  i-  C . SC2  A . SA2  -f  B sB®-{-  C • SCa 
SG  (A  -f-  B +-  c)  A . sa  -J-  B . si  -f  C . Sc 

by  prop.  42,  which  is  the  distance  of  the  centre  of  oscillation 
o,  below  the  axis  of  suspension  ; where  any  of  the  products 
a . sa,  b . s b,  must  be  negative,  when  a,  b,  &zc  lie  on  the 
other  side  of  s,  So  that  this  is  the  same  expression  as  that 
for  the  distance  of  the  centre  of  percussion,  found  in  prop.  54. 


Hence  it  appears,  that  the  centres  of  percussion  and  of  os- 
cillation, are  in  the  very  same  point.  And  therefore  the  prop- 
erties in  all  the  corollaries  there  found  for  the  former,  are  to 
be  here  understood  of  the  latter. 


290.  Corol.  1.  If  p be  any  particle  of  a body  b,  and  d its 
distance  from  the  axis  of  motion  s ; also  g,  o the  centres  of 
gravity  and  oscillation.  Then  the  distance  of  the  centre  of 
oscillation  of  the  body,  from  the  axis  of  motion,  is  - 

sum  of  all  th^  pa 2 
S0  = 7. 

SG  X the  body  o 


291.  Corol.  2.  If  b denote  the  matter  in  any  compound 
body,  whose  centres  of  gravity  and  oscillation  are  g and  o ; 
the  body  p,  which  being  placed  at  p,  where  the  force  acts  as 
in  the  last  proposition,  and  which  receives  tbe  same  motion 

from  that  force  as  the  compound  body  b,  is  p = . b. 


For,  by  corol.  2,  prop  54,  this  body  p is  = 

. SA2  4-  B . SB2  + c . sc2  . ' 

— ^ I — . But,  by  corol.  1,  prop.  53, 


198 


STATICS. 


sg  . so  . b = a . sa2  4-  b . sb2  -J"  c . sc3  . therefor  e 

^ SG  . SO 
SP 


SCHOLIUM. 


292.  By  the  method  of  Fluxions  ; the  centre  of  oscillation, 
for  a regular  bod}  , will  be  found  from  cor.  1.  But  for  an 
irregular  one  ; suspend  it  at  the  given  point  ; and  hang  up 
also  a simple  pendulum  of  such  a length,  that  making  them 
both  vibrate,  they  may  keep  time  together.  Then  the 
length  of  the  simple  pendulum,  is  equal  to  the  distance  of 
the  centre  of  oscillation  of  the  body,  below  the  point  of  sus- 
pension 


293.  Or  it  will  be  ill  better  found  thus  : Suspend  the 
body  very  freely  by  the  given  point,  and  make  it  vibrate  in 
small  arcs,  counting  the  number  of  vibrations  it  makes  in 
any  time,  as  a minute,  by  a good  stop  watch  ; and  let  that 
number  of  vibrations  made  in  a minute  be  called  n : Then 


shall  the  distance  of  the  centre  of  oscillation,  be  so  = 


140850 

nn 


inches.  For  the  length  of  the  pendulum  vibrating  seconds, 
or  60  times  in  a minute,  being  391  inches  ; and  the  lengths 
of  pendulums  being  reciprocally  as  the  square  of  the  number 
of  vibrations  made  in  the  same  time  ; therefore  - - - - 


ft2  : 603  : : 39j  : 


60s  X oH 140850 

n n nn 


the  length  of  the 


pendulum  which  vibrates  n times  in  a minute,  or  the  distance 
of  the  centre  of  oscillation  below  the  axis  of  motion. 


294.  The  foregoing  determination  of  the  point,  into  which 
all  the  matter  of  a body  being  collected,  it  shall  oscillate  in 
the  same  manner  as  before,  only  respects  the  case  in  which 
the  body  is  put  in  motion  by  the  gravity  of  its  own  particles, 
and  the  point  is  the  centre  of  oscillation  : but  when  the  body 
is  put  in  motion  by  some  other  extraneous  force,  instead  of 
its  gravity,  then  the  point  is  different  from  the  former,  and 
is  called  the  Centre  of  Gyration  ; which  is  determined  in  the 
following  manner  : 


PRO. 


CENTRE  OF  GYRATION. 


199 


PROPOSITION  LVn. 


295.  To  determine  the  Centre  of  Gyration  of  a Compound  Body 
or  of  a System  of  Bodies. 

Let  r be  the  centre  of  gyration,  or 
the  point  into  which  all  the  particles  a, 
b,  c,  &c.  being  collected,  it  shall  eceive 
the  same  angular  motion  from  a force  f 
acting  at  p,  as  the  whole  system  re- 
ceives. 

Now,  by  cor.  3.  pr.  54,  the  angular 
velocity  generated  in  the  system  by  the 

f S p 

force  f is  as — — , and 

J A . SA2  -f-  B . SB  2 &C. 

by  the  same,  the  angular  velocity  of  the  system  placed  in  r, 

■f  s p 

J ' • then,  by  making  these  two  expres- 


is 


(A  4-  B -f-  C &c.)  . sr2 
sions  equal  to  each  other,  the  equation  gives 
a . SA 2 4-  B . SB 2 + c sc 
= v/  — 


for  the  distance  of  the 


sr  v , 

A ”f-  B -f-  C 

centre  of  gyration  below  the  axis  of  motion. 

296.  Corol  1.  Because  a . sa2  -f-  b . sb2  &c.  = sg  . so  . b, 
where  g is  the  centre  of  gravity  o the  centre  of  oscillation, 
and  b the  body  A + B-f  c &c.  ; therefore  sr  = sg  . so  ; that 
is,  the  distance  ot  the  centre  of  gyration,  is  a mean  propor- 
tional between  those  of  gravity  and  oscillation. 

297.  Corol.  2.  If  p denote  any  particle  of  a body  b,  at  d dis- 


tance from  the  axis  of  motion  ; then  sr2 


sum  of  all  the  pa !2 

body  b 


PROPOSITION  LVIII. 

298.  To  determine  the  velocity  with  which  a Ball  moves , which 
being  shot  against  a Ballistic  Pendulum,  causes  it  to  vibrate 
through  a given  Angle. 

The  Ballistic  Pendulum  is  a heavy  block 
of  wood  mn,  suspended  vertically  by  a strong 
horizontal  iron  axis  at  s,  to  which  it  is  con- 
nected by  a firm  iron  stem.  This  problem 
is  the  application  of  the  last  proposition,  or 
of  prop.  54,  and  was  invented  by  the  very 
ingenious  Mr.  Piobins,  to  determine  the  ini- 
tial velocities  of  military  projectiles  ; a cir- 
cumstance very  useful  in  that  science  ; and 
it  is  the  best  method  yet  known  for  deter- 
mining them  with  any  degree  of  accuracy. 


Let 


200 


STATICS. 


Let  g,  r,  o be  the  centres  of  gravity,  gyration,  and  oscil- 
lation, as  determined  by  the  foregoing  propositions  ; and  let  r 
be  the  point  where  the  ball  strikes  the  face  of  the  pendulum  ; 
the  momentum  of  which,  or  the  product  of  its  weight  and 
velocity,  is  expressed  by  the  force/-,  acting 
at  p,  in  the  foregoing  propositions.  Now, 

Put  p — the  whole  weight  of  the  pendul. 
b = the  weight  of  the  ball, 
g = sg  the  dist.  of  the  cen.  of  grav. 

0 — so  the  dist.  of  the  cen.  of  oscilla. 
r = sr  = *ygo  the  dist.  of  cen.  of  gyr. 

1 = sp  the  dist.  of  the  point  of  impact, 
v — the  velocity  of  the  ball, 
u = that  of  the  point  of  impact  p, 
c = chord  of  the  arc  described  by  o. 


By  prop.  56,  if  the  mass  p be  placed  all  at  r,  the  pen- 
dulum will  receive  the  same  motion  from  the  blow  in  the 

point  p : and  as  sp2  : sr2  : : p : p or-/  p or  —p, (prop. 54), 

the  mass  which  being  placed  at  p,  the  pendulum  will  still 
receive  the  same  motion  as  before.  Here  then  are  two 

quantities  of  matter,  namely,  b and— p,  the  former  moving 


with  the  velocity  v,  and  striking  the  latter  at  rest  ; to  deter- 
mine their  common  velocity  u,  with  which  they  will  jointly 
proceed  forward  together  after  the  stroke.  In  which  case, 
by  the  law  of  the  impact  of  non-elastic  bodies  we  have 

— p H-  b : b : : v : u,  and  therefore  v — ^ u the  velo- 
ii  bii 

city  of  the  ball  in  terms  of  u,  the  velocity  of  the  point  p,  and 
the  known  dimensions  and  weights  of  the  bodies. 


But  now  to  determine  the  value  of  it,  we  must  have  re- 
course to  the  angle  through  which  the  pendulum  vibrates  ; 
for  when  the  pendulum  descends  down  again  to  the  vertical 
position,  it  will  have  acquired  the  same  velocit)'  with  which 
it  began  to  ascend,  and,  by  the  laws  of  falling  bodies,  the 
velocity  of  the  centre  of  oscillation  is  such,  as  a heavy  body 
would  acquire  by  freely  falling  through  the  versed  sine  of 
the  arc  described  by  the  same  centre  o But  the  chord  of 
that  arc  is  c,  and  its  radius  is  o ; and,  by  the  nature  of  the 
circle,  the  chord  is  a mean  proportional  between  the  versed 

sine  and  diameter,  therefore  2 o : c : : c the  versed  sine 

2o 

of  the  arc  described  by  o.  Then,  by  the  laws  offalling  bodies 

v/  16 


HYDROSTATICS. 


201 


qq 

v/  16XV  : \ 32i  : c y/  — , the  velocity  acquired  by  the 

point  o in  descending  through  the  arc  whose  chord  is  c, 

where  a = 16xlj  feet  : and  therefore  o : i : : c : — */— , 

0 0 0 

which  is  the  velocity  w,  of  the  point  p. 

Then,  by  substituting  this  value  for  u,  the  velocity  of  the 

ball  before  found,  becomes  v = ^ X c */  — . So  that 

the  velocity  of  the  ball  is  directly  as  the  chord  of  the  arc  de- 
scribed by  the  pendulum  in  its  vibration. 

SCHOLIUM. 

299.  In  the  foregoing  solution,  the  change  in  the  centre 
of  oscillation  is  omitted,  which  is  caused  by  the  ball  lodging 
in  the  point  p.  But  the  allowance  for  that  small  change,  and 
that  of  some  other  small  quantities,  may  be  seen  in  my  Tracts, 
where  all  the  circumstances  of  this  method  are  treated  at  full 
length. 

300.  For  an  example  in  numbers  of  this  method,  suppose 
the  weights  and  dimensions  to  be  as  follow  : namely, 

p = 5701b,  Then 

b = 18oz.  l|dr.  bii+gop  _l-13l  X94-32+78iX84£x570 

= M3Hb,  —fa  1-131  X 94^X841  ’ 


% = 781  inc. 
j.»  = 841  jnc. 

= 7-065  feet 
= 94X3X  inc. 
c = 18-73  inc. 


18-73 

12 


= 656-56, 


, , ,2 a , 32i  193 

And  y/—~y/ L —y/ 

v o v 7-065  v 42-39 


2-1337. 


Therefore  656-56  X 2-1337  or  1401  feet,  is  the  velocity,  per 
i second,  with  which  the  ball  moved  when  it  struck  the  pendu- 
lum. 


OF  HYDROSTATICS. 


301.  Hydrostatics  is  the  science  which  treats  of  the  pres- 
ire,  or  weight,  and  equilibrium  of  water  and  other  fluids,  es- 
ecially  those  that  are  non-elastic. 

302.  A fluid  is  elastic,  when  it  can  be  reduced  into  a less 
dume  by  compression,  and  which  restores  itself  to  its  former 
ilk  again  when  the  pressure  is  removed  ; as  air.  And  it  is 
n-elastic,  when  it  is  not  comj:  ressible  by  such  force  ; as 
iter,  &c. 

Vol.  II.  27 


PRO- 


202 


HYDROSTATICS. 


PROPOSITION  L1X, 

303.  If  any  Part  of  a Fluid  be  raised  higher  than  the  rest,  by 
any  Force,  and  then  left  to  itself ; the  higher  Parts  will  descend 
to  the  lower  Places,  and  the  Fluid  will  not  rest,  till  its  Surface 
be  quite  even  and  level. 

For,  the  parts  of  a fluid  being  easily  moveable  every  way, 
the  higher  parts  will  descend  by  their  superior  gravity,  and 
raise  the  lower  parts,  till  the  whole  come  to  rest  in  a level  or 
horizontal  plane. 

304.  Corol.  1.  Hence,  water  that  com- 
municates with  other  water,  by  means  of 
a close  canal  or  pipe,  will  stand  at  the  same 
height  in  both  places  Like  as  water  in 
the  two  legs  of  a syphon. 

305.  Corol.  2.  For  the  same  reason,  if 
a fluid  gravitate  towards  a centre  ; it  will 
dispose  itself  into  a spherical  figure,  the 
centre  of  which  is  the  centre  of  force. 

Like  the  sea  in  respect  of  the  earth. 

PROPOSITION  LX. 

306.  When  a Fluid  is  at  Rest  in  a Vessel,  the  Base  of  which  is 
Parallel  to  the  Horizon  ; Equal  Parts  of  the  Base  are  Equally 
Pressed  by  the  Fluid. 

For,  on  every  equal  part  of  this  base  there  is  an  equal 
column  of  the  fluid  supported  by  it.  And  as  all  the  columns 
are  of  equal  height,  by  the  last  proposition  they  are  of  equal 
weight,  and  therefore  they  press  the  base  equally  ; that  is, 
equal  parts  of  the  base  sustain  an  equal  pressure. 

_ 9 * 

307.  Corol.  1.  All  parts  of  the  fluid  press  equally  at  the 
same  depth.  For,  if  a plane  parallel  to  the  horizon  be  con- 
ceived to  be  drawn  at  that  depth  : then  the  pressure  being 
the  same  in  any  part  of  that  plane,  by  the  proposition,  there- 
fore the  parts  of  the  fluid,  instead  of  the  plane,  sustain  the 
same  pressure  at  the  same  depth. 

308.  Corol.  2.  The  pressure  of  the  fluid  at  any  depth,  is 
as  the  depth  of  the  fluid.  For  the  pressure  is  as  the  weight, 
and  the  weight  is  as  the  height  of  the  fluid. 


309.  Corol 


PRESSURE  OF  FLUIDS. 


203 


309.  Corol.  5.  The  pressure  of  the  fluid  on  any  horizontal 
surface  or  plane,  is  equal  to  the  weight  of  a column  of  the 
fluid,  whose  base  is  equal  to  that  plane,  and  altitude  is  its 
depth  below  the  upper  surface  of  the  fluid. 

PROPOSITION  LXI. 

310.  When  a Fluid  is  Pressed  by  its  own  Weight,  or  by  any 
other  Force  ; at  any  Point  it  Presses  Equally,  in  all  Direc- 
tions whatever. 

This  arises  from  the  nature  of  fluidity,  by  which  it  yields 
to  any  force  in  any  direction.  If  it  cannot  recede  from  any 
force  applied,  it  will  press  against  other  parts  of  the  fluid  in 
the  direction  of  that  force.  And  the  pressure  in  all  direc- 
tions will  be  the  same  : for  if  it  were  less  in  any  part,  the 
fluid  would  move  that  way,  till  the  pressure  be  equal  every 
way. 

311.  Gorol.  1.  In  a vessel  containing  a fluid  ; the  pressure 
is  the  same  against  the  bottom,  as  against  the  sides,  or  even 
upwards  at  the  same  depth. 

312.  Corol.  2.  Hence,  and  from 
the  last  proposition,  if  abcd  be  a 
vessel  of  water,  and  there  be  taken, 
in  the  base  produced,  de,  to  repre- 
sent the  pressure  at  the  bottom  ; 
joining  ae,  and  drawing  any  pa- 
rallels to  the  base,  as  fg,  hi  ; then 
shall  fg  represent  the  pressure  at 
the  depth  ag,  and  hi  the  pressure  at  the  depth  ai,  and  so 
on  ; because  the  parallels  - fg,  hi,  ed, 

by  sim.  triangles  are  as  the  depths  ag,  ai,  ad  : 
which  are  as  the  pressures,  by  the  proposition. 

And  hence  the  sum  of  all  the  fg,  hi,  &c.  or  area  of  the 
triangle  ade,  is  as  the  pressure  against  all  the  points  g,  i, 
&c.  that  is,  against  the  line  ad.  But  as  every  point  in  the 
line  cd  is  pressed  with  a force  as  de,  and  that  thepce  the 
pressure  on  the  whole  line  cd  is  as  the  rectangle  ed  . nc, 
while  that  against  the  side  is  as  the  triangle  ade  or  ^ad  . de  ; 
therefore  the  pressure  on  the  horizontal  line  dc,  is  to  the 
pressure  against  the  vertical  line  da,  as  dc  to  4-da.  And 
hence,  if  the  vessel  be  an  upright  rectangular  one,  the  pres- 
sure on  the  bottom,  or  whole  weight  of  the  fluid,  is  to  the 
pressure  against  one  side,  as  the  base  is  to  half  that  side. 
Therefore  the  weight  of  the  fluid  is  to  the  pressure  against 

all 


204  HYDROSTATICS, 

all  the  four  upright  sides,  as  the  base  is  to  half  the  upright 
surface.  And  the  same  holds  true  also  in  any  upright  vessel, 
whatever  the  sides  be,  or  in  a cylindrical  vessel.  Or  in  the 
cylinder,  the  weight  of  the  fluid,  is  to  the  pressure  against 
the  upright  surface,  as  the  radius  of  the  base  is  to  double  the 
altitude. 

Also,  when  the'rectangular  prism  becomes  a cube,  it  appears 
that  the  weight  of  the  fluid  on  the  base,  is  double  the  pres- 
sure against  one  of  the  upright  sides,  or  half  the  pressure 
against  the  whole  upright  surface. 

313.  Carol.  3.  The  pressure  of  a fluid  against  any  upright 
surface,  as  the  gate  of  a sluice  or  canal,  is  equal  to  half  the 
weight  of  a column  of  the  fluid  whose  base  is  equal  to  the 
surface  pressed,  and  its  altitude  the  same  as  the  altitude  of 
that  surface.  For  the  pressure  on  a horizontal  base  equal 
to  the  upright  surface,  is  equal  to  that  column  ; and  the  pres- 
sure on  the  upright  surface,  is  but  half  that  on  the  base,  of  the 
same  area. 

So  that*  if  b denote  the  breadth,  and  d the  depth  of  such 
a gate  or  upright  surface  ; then  the  pressure  against  it,  is 
equal  to  the  weight  of  the  fluid  whose  magnitude  is  ±bd2  == 
J-ab  . ad2.  Hence,  if  the  fluid  be  water,  a cubic  foot  of 
which  weighs  1000  ounces,  or  624  pounds  ; apd  if  the  depth 
ad  be  12  feet,  the  breadth  ab  20  feet  ; then  the  content,  or 
Iab  . ad2,  is  1440  feet  ; and  the  pressure  is  1440000  ounces, 
or  90000  pounds,  or  40^  tons. 

PROPOSITION  LXII. 

314.  The  pressure  of  a Fluid  on  a Surface  any  how  immersed 
in  it,  either  Perpendicular r or  Horizontal,  or  Oblique  ; is 
Equal  to  the  Weight  of  a Column  of  the  Fluid,  whose  Base  is 
equal  to  the  Surface  pressed,  and  its  Altitude  equal  to  the 
Depth  of  the  Centre  of  Gravity  of  the  Surface  pressed  below 
the  Top  or  Surface  of  the  Fluid. 

For,  conceive  the  surface  pressed  to  be  divided  into  innu- 
merable sections  parallel  to  the  horizon  ; and  let  s denote 
any  one  of  those  horizontal  sections,  also  d its  distance  or 
depth  below  the  top  surface  of  the  fluid.  Then,  by  art.  309, 
the  pressure  of  the  fluid  on  the  section  is  equal  to  the  weight 
of  ds  ; consequently  the  total  pressure  on  the  whole  surface 
is  equal  to  all  the  weights  ds.  But,  if  b denote  the  whole 
surface  pressed,  aod  g the  depth  of  its  centre  of  gravity  be- 
low the  top  of  the  fluid  ; then,  by  art.  256  or  259,  bg  is  equal 


PRESSURE  OF  FLUIDS. 


205 


to  the  sum  of  all  the  ds.  Consequently  the  whole  pressure 
of  the  fluid  on  the  body  or  surface  b is  equal  to  the  weight 
of  the  bulk  bg  of  the  fluid,  that  is,  of  the  column  whose  base 
is  the  given  surface  b,  and  its  height  is  g the  depth  of  the 
centre  of  gravity  in  the  fluid. 


PROPOSITION  LXm. 

315.  The  Pressure  of  a Fluid,  on  the  Base  of  the  Vessel  in  which 

it  is  contained,  is  as  the  Base  and  Perpendicular  Altitude  ; 

whatever  be  the  Figure  of  the  Vessel  that  contains  it. 

If  the  sides  of  the  base  be  upright,  so  that 
it  be  a prism  of  a uniform  width  throughout  ; 
then  the  case  is  evident ; for  then  the  base 
supports  the  whole  fluid,  and  the  pressure  is 
just  equal  to  the  weight  of  the  fluid. 

But  if  the  vessel  be  wider'at  top  than  bot- 
tom ; then  the  bottom  sustains  or  is  pressed 
by,  only  the  part  contained  within  the  up- 
right lines  ac,  bD  ; because  the  parts  Aca, 

BDb  are  supported  by  the  sides  ac,  bd  ; 
and  those  parts  have  no  other  effect  on  the 
part  abDC  than  keeping  it  in  its  position,  by 
the  lateral  pressure  against  ac  and  bD,  which 
does  not  alter  its  perpendicular  pressure  downwards.  And 
thus  the  pressure  on  the  bottom  is  less  than  the  weight  of 
the  contained  fluid. 

And  if  the  vessel  be  widest  at  bottom  ; then 
the  bottom  is  still  pressed  with  a weight  which 
is  equal  to  that  of  the  whole  upright  column 
abDC.  For,  as  the  parts  of  the  fluid  are  in 
equilibrio,  all  the  parts  have  an  equal  pressure 
at  the  same  depth  ; so  that  the  parts  within  cc 
and  dn  press  equally  as  those  in  cd,  and  there- 
fore_  equally  the  same  as  if  the  sides  of  the  vessel  had  gone 
upright  to  a and  b,  the  defect  of  fluid  in  the  parts  Aca 
and  bd b being  exactly  compensated  by  the  downward  pres- 
sure or  resistance  of  the  sides  ac  and  bd  against  the  con- 
tiguous fluid  And  thus  the  pressure  on  the,  base  may  be 
made  to  exceed  the  weight  of  the  contained  fluid,  in  any  pro- 
portion whatever. 

So  that,  in  general,  be  the  vessels  of  any  figure  whatever, 
regular  or  irregular,  upright  or  sloping,  or  variously  wide 
and  narrow  in  different  parts,  if  the  bases  and  perpendicular 
altitudes  be  but  equal,  the  bases  always  sustain  the  same 
pressure.  -And  as  that  pressure,  in  the  regular  upright 

vessel. 


a AB 


206 


HYDROSTATICS. 


vessel,  is  the  whole  column  of  the  fluid,  which  is  as  the  base 
and  altitude  ; therefore  the  pressure  in  all  figures  is  in  that 
same  ratio 

316.  Corol.  1 Hence,  when  the  heights  are  equal,  the 
pressures  are  as  the  bases.  And  when  the  bases  are  equal, 
the  pressure  is  as  the  height.  But  when  both  the  heights 
and  bases  are  equal,  the  pressures  are  equal  in  all,  though 
their  contents  be  ever  so  different 

317  Corol.  2.  The  pressure  on  the  base  of  any  vessel,  is 
the  same  as  on  that  of  a cylinder,  of  an  equal  base  and  height. 

318  Corol.  3.  If  there  be  an  inverted  sy- 
phon, or  bent  tube,  abc,  containing  two  dif- 
ferent fluids  cd,  abd,  that  balance  each  other 
or  rest  in  equilibrio  ; then  their  heights  in 
the  two  legs,  ae,  cd,  above  the  point  of  meet- 
ing will  be  reciprocally  as  their  densities. 

For,  if  they  do  not  meet  at  the  bottom, 
the  part  bd  balances  the  part  be,  and  there- 
fore the  part  cd  balances  the  part  ae  ; that 
is,  the  weight  of  cd  is  equal  to  the  weight 
of  ae.  And  as  the  surface  at  d is  the  same 
where  they  act  against  each  other,  therefore 
ae  : cd  : : density  of  cd  : density  of  ae 

So,  if  cd  be  water,  and  ae  quicksilver,  which  is  near  14 
times  heavier;  then  cd  will  be  = 14ae  ; that  is,  if  ae  be 


l inch,  cd  will  be  14  inches 
inches  ; and  so  on. 


if  ae  be  2 inches,  cd  will  be  28 


PROPOSITION  LXIV. 

319.  If  a Body  be  Immersed  in  a Fluid  of  the  same  Density 
or  Specific  Gravity ; it  ■will  Rest  in  any  Place  where  it  is  put. 
But  a Body  of  Greater  Density  will  Sink  ; and  one  of  a Less 
Density  will  Rise  to  the  Top,  and  Float. 


The  body,  being  of  the  same  den- 
sity, or  of  the  same  weight  with  the 
like  bulk  of  the  fluid,  will  press  the 
fluid  under  it,  just  as  much  as  if  its 
space  was  filled  with  the  fluid  itself. 
The  pressure  then  ail  around  it  will 
be  the  same  as  if  the  fluid  were  in 
its  place  ; consequently  there  is  no 
force,  neither  upward  nor  down- 
ward, to  put  the  body  out  of  its 
place.  And  therefore  it  will  remain 
w’herever  it  is  put. 


But 


PRESSURE  OF  FLUIDS. 


207 


But  if  the  body  be  lighter  ; its  pressure  downward  will  be 
less  than  before,  and  less  than  the  water  upward  at  the  same 
depth  ; therefore  the  great  force  will  overcome  the  less,  and 
push  the  body  upward  to  a. 

And  if  the  body  be  heavier  than  the  fluid,  the  pressure 
downward  will  be  greater  than  the  fluid  at  the  same  depth  ; 
therefore  the  greater  force  will  prevail,  and  carry  the  body 
down  to  the  bottom  at  c. 

320.  Corol.  1.  A body  immersed  in  a fluid,  loses  as  much 
weight,  as  an  equal  bulk  of  the  fluid  weighs.  And  the  fluid 
gains  the  same  weight.  Thus,  if  the  body  be  of  equal  densi- 
ty with  the  fluid,  it  loses  all  its  weight,  and  so  requires  no 
force  but  the  fluid  to  sustain  it  If  it  be  heavier,  its  weight 
in  the  water  will  be  only  the  difference  between  its  own  weight 
and  the  weight  of  the  same  bulk  of  water  ; and  it  requires  a 
force  to  sustain  it  just  equal  to  that  difference  But  if  it  be 
lighter,  it  requires  a force  equal  to  the  same  difference  of 
weights  to  keep  it  from  rising  up  in  the  fluid. 

321.  Corol.  2.  The  weights  lost,  by  immerging  the  same 
body  in  different  fluids,  are  as  the  specific  gravities  of  the 
fluids.  And  bodies  of  equal  weight,  but  different  bulks,  lose, 
in  the  same  fluid,  weights  which  are  reciprocally  as  the  spe- 
cific gravities  of  the  bodies,  or  directly  as  their  bulks. 

322.  Corol.  3.  The  whole  weight  of  a body  which  will  float 
in  a fluid,  is  equal  to  as  much  of  the ‘fluid,  as  the  immersed 
part  of  the  body  takes  up,  when  it  floats.  For  the  pressure 
under  the  floating  body,  is  just  the  same  as  so  much  of  the 
fluid  as  is  equal  to  the  immersed  part  ; and  therefore  the 
weights  are  the  same. 

323.  Corol.  4.  Hence  the  magnitude  of  the  whole  body,  is 
to  the  magnitude  of  the  part  immersed,  as  the  specific  gravity 
of  the  fluid,  is  to  that  of  the  body.  For,  in  bodies  of  equal 
weight,  the  densities,  or  specific  gravities,  are  reciprocally  as 
their  magnitudes. 

324.  Corol.  5.  And  because  when  the  weight  of  a body 
taken  in  a fluid,  is  subtracted  from  its  weight  out  of  the  fluid, 
the  difference  is  the  weight  of  an  equal  bulk  of  the  fluid  ; this 
therefore  is  to  its  weight  in  the  air,  as  the  specific  gravity  of 
the  fluid,  is  to  that  of  body. 

Therefore,  if  w be  the  weight  of  a body  in  air, 
w its  weight  in  water,  or  any  fluid, 
s the  specific  gravity  of  the  body,  and 
* the  specific  gravity  of  the  fluid  ,• 


then 


soa 


hydrostatics: 


then  w — w : w.:  : s : s,  which  proportion  will  give  either  of 
those  specific  gravities,  the  one  from  the  other. 

W 

Thus  s = - — — s , the  specific  gravity  of  the  body  ; 

w — 

and  s = — — — s,  the  specific  gravity  of  the  fluid. 

So  that  the  specific  gravities  of  bodies,  are  as  their  weights 
in  the  air  directly,  and  their  loss  in  the  same  fluid  inversely. 


325.  Corol.  6.  And  hence,  for  two  bodies  connected  togeth- 
er, or  mixed  together  into  one  compound,  of  different  speci- 
fic gravities,  we  have  the  following  equations,  denoting  their 
weights  and  specific  gravities,  as  below,  viz. 


h = weight  of  the  heavier  body  in  air, 
ft  sss  weight  of  the  same  in  water, 
l = weight  of  the  lighter  body  in  air, 
l — weight  of  the  same  in  water, 
c = weight  of  the  compound  in  air, 
c = weight  of  the  same  in  water, 
w = the  specific  gravity  of  water.  Then, 


s its  spec,  gravity  ; 


| s its  spec,  gravity  ; 
>/its  spec,  gravity  ; 


1st,  (H  — ft)  S = HZ®, 

2d,  (l  — l)  s = lw, 

3d,  (c  — c)  f = cw, 

4th,  h + l — c, 

5th,  ft  -f*  l — c, 

H L C 

6th, 1 — — 

s s f dividing  the  absolute  weight  of  the 
body  by  its  loss  in  water,  and  multiplying  by  the  specific  gra- 
vity of  water. 


From  which  equations  may  be  found 
any  of  the  above  quantities,  in  terms  of 
the  rest. 

Thus,  from  one  of  the  first  three 
equations,  is  found  the  specific  gra- 
vity of  any  body,  as  s = by 


But  if  the  body  l be  lighter  than  water  ; then  l will  be 
negative,  and  we  must  divide  by  l l instead  of  l — l,  and 
to  find  l we  must  have  recourse  to  the  compound  mass  c ; and 
because,  from  the  4th  and  5th  equations,  l — l-c-c  — 

h— ft,  therefore  s = — ; that  is,  divide  the 

(c-c)-(h-A)’ 

absolute  weight  of  the  light  body,  by  the  difference  be- 
tween the  losses  in  water,  of  the  compound  and  heavier  body, 
and  multiply  by  the  specific  gravity  of  water.  Or  thus, 

S f 

s = -,  as  found  from  the  last  equation. 

c s ^ h/ 

Also,  if  it  were  required  to  find  the  quantities  of  two  ingre- 
dients mixed  in  a compound,  the  4th  and  6th  equations  would 
give  their  values  as  follows,  viz. 

h •= 


SPECIFIC  GRAVITY. 


209 


(J'—S>S  , li- 
lt s=  J- c,  and  l — — 

(s — S)  (s 


-*)/ 


tC, 


the  quantities  of  the  two  ingredients  h 
pound  c.  And  so  for  any  other  demand 


and  l,  in  the  com- 


PROPOSITION  LXV. 

To  find  the  Specific  Gravity  of  a Body . 

326.  Case  i. — When  the  body  is  heavier  than  water  : weigh 
it  both  in  water  and  out  of  water,  and  take  the  difference, 
which  will  be  the  weight  lost  in  water.  Then,  by  corol.  6, 

prop.  64,  s = BI1; - , where  b is  the  weight  of  the  body  out 

of  water,  b its  weight  in  water,  s its  specific  gravity,  and  w 
the  specific  gravity  of  water.  That  is, 

As  the  weight  lost  in  water, 

Is  to  the  whole  or  absolute  weight, 

So  is  the  specific  gravity  of  water, 

To  the  specific  gravity  of  the  body. 

Example.  If  a piece  of  stone  weigh  10  lb,  but  in  water 
only  6£  lb,  required  its  specific  gravity,  that  of  water  being 
1000  ? Ans.  3077. 

327.  Case  ii. — When  the  body  is  lighter  than  water,  so  that 
• t will  not  sink  : annex  to  it  a piece  of  another  body,  heavier 
:han  water,  so  that  the  mass  compounded  of  the  two  may 
;ink  together.  Weigh  the  denser  body  and  the  compound 
nass,  separately,  both  in  water  and  out  of  it  ; then  find  how 
such  each  loses  in  water,  by  subtracting  its  weight  in  water 
rom  its  weight  in  air  ; and  subtract  the  less  of  these  re- 
nainders  from  the  greater.  Then  say,  by  proportion, 

As  the  last  remainder, 

I Is  to  the  weight  of  the  light  body  in  air, 

So  is  the  specific  gravity  of  water, 

To  the  specific  gravity  of  the  body. 

That  is,  the  specific  gravity  is  s — 

y cor.  6,  prop.  64. 


(c  _ c)  -(h  ■-  h.y 


Example.  Suppose  a piece  of  elm  weighs  15  lb  in  air  ; 
nd  that  a piece  of  copper,  which  weighs  18  lb  in  air  and 
3 lb  in  water,  is  affixed  to  it,  and  that  the  compound  weighs 
lb  in  water  ; required  the  specific  gravity  of  the  elm  ? 

Ans.  600. 


Vor..  II. 


28 


328.  Case 


21© 


HYDROSTATICS. 


328.  Case  iii. — For  a fluid  of  any  sort. — Take  a piece  of 
a .body  of  known  specific  gravity  ; weigh  it  both  in  and  out 
of  the  fluid,  finding  the  loss  of  weight  by  taking  the  differ- 
ence of  the  two  ; then  say, 


As  the  whole  or  absolute  weight, 

Is  to  the  loss  of  weight. 

So  is  the  specific  gravity  of  the  solid, 
To  the  specific  gravity  of  the  fluid. 


B—  b 


That  is,  the  spec.  gray,  w = s,  by  cor.  6,  pr.  64. 

Example.  A piece  of  cast  iron  weighed  35T6J«  ounce® 
in  a fluid,  and  40  ounces  out  of  it  ; of  what  specific  gravity  is 
that  fluid  ? Ans.  1000.il 


PROPOSITION  LXVI. 


329.  To  find  the  Quantities  of  Two  Ingredients  in  a Given 
Compound. 

Take  the  three  differences  of  every  pair  of  the  three  spe- 
cific gravities,  namely,  the  specific  gravities  of  the  compound 
and  each  ingredient  ; and  multiply  each  specific  gravity  by 
the  difference  of  the  other  two.  Then  say,  by  proportion, 

As  the  greatest  product, 

Is  to  the  whole  weight  of  the  compound, 

So  is  each  of  the  other  two  products, 

To  the  weights  of  the  two  ingredients. 

That  is,  h = ■ -^4 c = the  one,  and  l = — — the 

(s-oy  (s— f)/ 

other,  by  cor.  6,  prop.  64. 


Example.  A composition  of  112  lb  being  made  of  tin 
and  copper,  whose  specific  gravity  is  found  to  be  8784  ; re- 
quired the  quantity  of  each  ingredient,  the  specific  gravity 
of  tin  oeing  7320.  and  that  of  copper  9000  1 
Answer,  chore  is  100  lb  of  copper,  ) - ,, 

and  consequently  12  lb  of  tin,  ' ln  e 


composition 


SCHOLIUM. 

330.  The  specific  gravities  of  several  sorts  of  matter,  a: 
found  from  experiments,  ire  expressed  by  the  numbers  an 
nexed  to  their  names  in  the  following  Table  : 

A TabU 


SPECIFIC  GRAVITY. 


211 


J1  Table  of  Specific  Gravities  of  Bodies. 


Platina  (pure)  - - - 

Fine  gold  - - - - 

Standard  gold  - - - 

Quicksilver  (pure)  - - 

Quicksilver  (common)  - 

Lead 

Fine  silver  - - - • 

Standard  silver  - - - 

Copper  ----- 
Copper  halfpence  - • 

Gun  metal  - - - - 

Cast  brass  - 

Steel  - 

Iron 

Cast  Iron  - - - - 

Tin 

Clear  crystal  glass  - - 

Granite 

Marble  and  hard  stone 
Common  green  glass  - 

Flint 

Common  stone  - 

331.  Note.  The  several  sorts  of  wood  are  supposed  to  be 
dry.  Also,  as  a cubic  foot  of  water  weighs  just  1000  ounces 
avoirdupois,  the  numbers  in  this  table  express  not  only  the 
specific  gravities  of  the  several  bodies,  but  also  the  weight  of 
a cubic  foot  of  each,  in  avoirdupois  ounces  ; and  therefore, 
by  proportion,  the  weight  of  any  other  quantity,  or  the 
quantity  of  any  other  weight,  may  be  known,  as  in  the  next 
two  propositions. 


230001  Clay 2160 

19400jBrick 2000 

17724|Common  earth  - - - 1984 

14000  Nitre 1900 

13600.  Ivory 1825 

1 l325!Brims'cone 1810 

1 lu9 1 Solid  gunpowder  - - - 1745 

10635jSand 1520 

9000!  Coal 12.50 

89 15!  Box -wood  - 1030 

8784|Sea-water 1030 

8000jCommon-water  - - - 1000 

7850|0ak 925 

7645;G’inpowder,  close  shaken  937 
7425|Ditto,  in  a loose  heap  - 836 

7320,  Ash 800 

3150  Maple  - 755 

3000  Elm  -.----  600 

2700  Fir 550 

2600  Charcoal  - - - 

2570  Cork 240 

2520|Air  at  a mean  state  - - 1| 


PROPOSITION  LXVII. 


332.  To  find  the  Magnitude  of  any  Body,  from  its  Weight. 

As  the  tabular  specific  gravity  of  the  body, 

Is  to  its  weight  in  avoirdupois  ounces, 

So  is  one  cubic  foot,  or  1728  cubic  inches, 

To  its  content  in  feet,  or  inches,  respectively. 

Example  1.  Required  the  content  of  an  irregular  block  of 
common  stone,  which  weighs  1 cwt.  or  1121b  ? 

Ans.  1228|JU-§  cubic  inches. 

Example  2.  How  many  cubic  inches  of  gunpowder  are 
there  in  1 lb  weight  ? Ans.  29^  cubic  inches  nearly. 

Example  3 


2 12 


HYDRAULICS. 


Example  3.  How  many  cubic  feet  are  there  in  a ton  weight 
of  dry  oak  ? Ans.  38i|4  cubic  feet. 

PROPOSITION  LXVI1I. 

333.  To  find  the  Weight  of  a Body  from  its  Magnitude. 

As  one  cubic  foot,  or  1728  cubic  inches, 

Is  to  the  content  of  the  body, 

So  is  the  tabular  specific  gravity, 

To  the  weight  of  the  body. 

Example  i . Required  the  weight  of  a block  of  marble, 
whose  length  is  03  feet,  and  breadth  and  thickness  each 
12  feet  ; ceing  the  dimensions  of  one  of  the  stones  in  the 
walls  of  balueck  ? 

Ans.  683t%  ton,  which  is  nearly  equal  to  the  burden  of 
an  East-lndia  ship. 

Example  2.  What  is  the  weight  of  1 pint,  ale  measure,  of 
gunrowder  ? Ans.  19  oz  nearly. 

Example  3.  What  is  the  weight  ot  a block  ot  dry  oak. 
which  measures  10  feet  in  length,  3 feet  broad,  and  2^  feet 
deep  or  thick  ? - Ans.  4335fflb 

U'  ■ _ ' ■ ■ ■'  • , . ■ j 

OF  HYDRAULICS. 

334.  Hydraulics  is  the  science  which  treats  of  the  mo- 
tion of  fluids,  and  the  forces  with  which  they  act  upon 
bodies. 

PROPOSITION  LXIX. 

335.  If  a Fluid  Run  through  a Canal  or  River,  or  Pipe  of  va- 
rious Widths,  always  filing  it  ; the  Velocity  of  the  Fluid  in 
different  Parts  of  it  ab,  cd,  will  be  reciprocally  as  the  Trans- 
verse Sections  in  those  Parts. 

That  is  veloc.  at  a : veloc. 
at  c : • cd  : ab  ; where  ab  and 
cd  denote,  not  the  diameters 
at  a and  b,  but  the  areas  or 
sections  there. 

For,  as  the  channel  is  always  equally  full,  the  quantity  of 
water  running  through  ab  is  equal  to  the  quantity  running 
through  cd,  in  the  same  time  ; that  is  the  column  through 


SPOUTING  OF  FLUIDS. 


213 


jib  is  equal  to  the  column  through  cd,  in  the  same  time  ; 
or  ab  X length  of  its  column  = cd  X length  of  its  column  ; 
therefore  ab  : cd  : : length  of  column  through  cd  : length  of 
column  through  ab.  But  the  uniform  velocity  of  the  water,  is 
as  the  space  run  over,  or  length  of  the  columns  ; therefore 
ab  : cd  : : velocity  through  cd  : velocity  through  ab. 

336.  Carol.  Hence,  by  observing  the  velocity  at  any  place 
ab,  the  quantity  of  water  discharged  in  a second,  or  any  other 
time,  will  be  found,  namely,  by  multiplying  the  section  ab  by 
the  velocity  there. 

But  if  the  channel  be  not  a close  pipe  or  tunnel,  kept 
always  full,  but  an  open  canal  cr  river  ; then  the  velocity  in 
all  parts  of  the  section  will  not  be  the  same,  because  the 
velocity  towards  the  bottom  and  sides  will  be  diminished  by 
the  friction  against  the  bed  or  channel,  and  therefore  a me- 
dium among  the  three  ought  to  be  taken.  So  if  the  velo- 
city at  the  top  be  - 100  feet  per  minute, 

that  at  the  bottom  - 60 

and  that  at  the  sides  - 50 


3)  210  sum  : 

dividing  their  sum  by  3 gives  70  for  the  mean  velocity,  which 
is  to  be  multiplied  by  the  section,  to  give  the  quantity  dis- 
charged in  a minute. 

PROPOSITION  LXX. 


337.  The  Velocity  with  which  a Fluid.  Runs  out  by  a Hole  in  the 
Bottom  or  Side  of  a Vessel,  is  Equal  to  that  which  is  Gene- 
rated by  Gravity  through  the  Height  of  the  Water  above  the 
Hole  ; that  is,  the  Velocity  of  a Heavy  Body  acquired  by  Fall- 
ing freely  through  the  Height  ab. 


Divide  the  altitude  ab  into  a great 
number  of  very  small  parts,  each  being  1, 
their  number  a,  or  a — the  altitude  ab. 

Now,  by  prop.  61,  the  pressure  of  the 
fluid  against  the  whole  b,  by  which  the 
motion  is  generated,  is  equal  to  the 
weight  of  the  column  of  fluid  above  it, 
that  is,  the  column  whose  height  is  ab 
or  a,  and  base  the  area  of  the  hole  b.  Therefore  the  pres- 
sure on  the  hole,  or  small  part  of  the  fluid  1,  is  to  its  weight, 
or  the  natural  force  of  gravity,  as  a to  1.  But,  by  art.  28, 
the  velocities  generated  in  the  same  body  in  any  time,  are  as 

those 


214 


HYDRAULICS. 


those  forces  ; and  because  gravity  generates  the  velocity  2 in 
descending  through  the  small  space  1,  therefore  1 : a . : 2 : 2a, 
the  velocity  generated  by  the  pressure  of  the  column  of  fluid 
in  the  same  time.  But  2 a is  also,  by  cord.  1,  prop.  6,  the 
velocity  generated  by  gravity  in  descending-  trough  a or  ab. 
That  is,  the  velocity  of  the  issuing  water,  is  equal  to  that  which 
is  acquired  by  a bod)  in  falling  through  the  height  ab. 

The  same  olhercsise. 

Because  the  momenta,  or  quantities  of  motion  generated 
in  two  given  bodies,  by  the  same  force,  acting  during  the 
same  or  an  equal  time,  are  equal.  And  as  the  force  in  this 
* case,  is  the  weight  of  the  superincumbent  column  of  the 
fluid  over  the  hole.  Let  the  one  body  to  be  moved,  be  that 
column  itself,  expressed  by  ah , where  a denotes  the  abilude 
ab,  and  h the  area  of  the  hole  ; and  the  other  body  is  the 
column  of  the  fluid  that  runs  out  uniformly  in  one  second 
suppose,  with  the  middle  or  medium  velocity  of  that  interval 
of  time,  which  is  -i hv . if  v be  the  whole  velocity  required. 
Then  the  mass  \hv , with  the  velocity  v.  gives  the  quantity 
of  motion  \liv  X v or  \hv2 . generated  in  one  second,  in  the 
spouting  water  : also  2 g,  or  32J-  feet,  is  the  velocity  generated 
in  the  mass  ah  during  the  same  interval  of  one  second  ; conse- 
quently ah  X 2 g,  or  2 ahg , is  tbe  motion  generated  in  the 
column  ah  in  the  same  time  of  one  second.  But  as  these 
two  momenta  must  be  equal,  this  gives  ±hv2  —■  2akg  : hence 

then  v 2 = 4ag,  and  v — 2^/ug,  for  the  value  of  the  velocity 
sought  : which  therefore  is  exactly  tne  same  as  the  velocity 
generated  by  the  gravity  in  falling  through  the  space  a,  or  the 
whole  height  of  the  fluid. 

For  example,  if  the  fluid  were  air,  of  the  whole  height  of 
the  atmosphere,  supposed  uniform,  which  is  about  5i  miles, 
or  27720  feet  = a.  Then  2^/ag  = 2 27720  X 16^  = 

1335  feet  = v the  velocity,  that  is,  the  velocity  with  which 
common  air  would  rush  into  a vacuum. 

338.  Corol.  1.  The  velocity,  and  quantity  run  out,  at  dif- 
ferent depths,  are  as  the  square  roots  of  the  depths.  For  the 
velocity  acquired  in  falling  through  ab,  is  as  ab. 

330.  Corol.  2.  The  fluid  spouts  out  with  the  same  velocity, 
whether  it  be  downward  or  upward,  or  sideways  ; because 
the  pressure  of  fluids  is  the  same  in  all  directions,  at  the 
same  depth.  And  therefore,  if  an  adjutage  be  turned  up- 
ward, the  jet  will  ascend,  to  the  height  of  the  surface  of  the 
water  in  the  vessel.  And  this  is  confirmed  by  experience, 
by  which  it  is  found  that  jets  really  ascend  nearly  to  the 

height 


SPOUTING  OF  FLUIDS. 


215 


height  of  the  reservoir,  abating  a small  quantity  only,  for  the 
friction  against  the  sides.  and  some  resistance  from  the  air  and 
from  the  oblique  motion  of  the  fluid  in  the  hole. 

3-V  Carol.  3 The  quantity  ran  out  in  any  time,  is  equal 
to  a column  or  prism,  whose  base  is  the  area  cf  the  hole,- and 
its  length  the  space  described  in  that  time  by  the  velocity  ac- 
quired by  falling  through  the  altitude  of  the  fluid.  And  the 
quantity  is  the  same,  whatever  be  the  figure  of  the  orifice,  if 
it  is  of  the  same  area. 

Therefore,  if  a denote  the  altitude  of  the  fluid, 
and  h the  area  of  the  orifice, 
also  g — 16tl  feet,  or  193  inches  ; 
then  2 h if  ag  will  be  the  quantity  of  water  discharged  in  a 
second  of  time  ; or  nearly  8^\kfa  cubic  feet,  when  a and  h 
are  taken  in  feet. 

So,  for  example,  if  the  height  a be  25  inches,  and  the  ori- 
fice h — 1 square  inch  ; then  2 hfag  ==  2v/25  XI 93  — 139 
cubic  inches,  which  is  the  quantity  that  would  be  discharged 
per  second. 

SCHOLIUM. 

341.  When  the  orifice  is  in  the  side  of  the  vessel,  then  the 
velocity  is  different  in  the  different  parts  of  the  hole,  being  less 
in  the  upper  parts  of  it  than  in  the  lower.  However,  when 
the  hole  is  but  small,  the  difference  is  inconsiderable,  and  the 
altitude  may  be  estimated  from  the  centre  of  the  hole  to  ob- 
tain the  mean  velocity.  But  when  the  orifice  is  pretty  large, 
then  the  mean  velocity  is  to  be  more  accurately  computed  by 
other  principles,  given  in  the  next  proposition. 

342.  It  is  not  to  be  expected  that  experiments,  as  to  the 
quantity  of  water  run  out,  will  exactly  agree  with  this  theory, 
both  on  account  of  the  resistance  of  the  air,  the  resistance  of 
the  water  against  the  sides  of  the  orifice,  and  the  oblique  mo- 
tion of  the  particles  the  water  in  entering  it.  For,  it  is  not 
merely  the  particles  situated  immediately  in  the  column  over 
the  hole,  which  enter  it  and  issue  forth,  as  if  that  column  only 
were  in  motion  ; but  also  particles  from  all  the  surrounding 
parts  of  the  fluid,  which  is  in  a commotion  quite  around  ; and 
the  particles  thus  entering  the  hole  in  all  directions,  strike 
against  each  other,  and  impede  one  another’s  motion  : from 
which  it  happens,  that  it  is  the  particles  in  the  centre  of  the 
hole  only  that  issue  out  with  the  whole  velocity  due  to  the  en- 
tire height  of  the  fluid,  while  the  other  particles  towards  the 
sides  of  the  orifices  pass  out  with  decreased  volocities  ; and 
hence  the  medium  velocity  through  the  orifice,  is  somewhat 
less  than  that  of  a single  body  only,  urged  with  the  same  pres- 
sure of  the  superincumbent  column  of  the  fluid.  And  experi- 
ments 


216 


HYDRAULICS. 


ments  on  the  quantity  of  water  discharged  through  apertures 
show  that  the  quantity  must  be  diminished,  by  those  causes 
rather  more  than  the  fourth  part  when  the  orifice  is  small,  or 
such  as  to  make  the  mean  velocity  nearly  equal  to  that  in  a 
body  falling  through  i the  height  of  the  fluid  above  the  ori- 
fice. 

343.  Experiments  have  also  been  made  on  the  extent  to 
which  the  spout  of  water  ranges  on  a horizontal  plane,  and 
compared  with  the  theory,  by  calculating  it  as  a projectile 
discharged  with  the  velocity  acquired  by  descending  through 
the  height  of  the  fluid.  For,  when  the  aperture  is  in  the 
side  of  the  vessel,  the  fluid  spouts  out  horizontally  with  a 
uniform  velocity,  which  combined  with  the  perpendicular 
velocity  from  the  action  of  gravity,  causes  the  jet  to  form 
the  curve  of  a parabola.  Then 
the  distances  to  which  the  jet  will 
spout  on  the  horizontal  plane  bg, 
will  be  as  the  roots  of  the  rect- 
angles of  the  segments  ac  . cb, 
ad  . db,  ae  . eb.  For  the  spaces 
bf,  bg,  are  as  the  times  and  hori- 
zontal velocities  ; but  the  velocity 
is  as  ac  ; and  the  time  of  the 
fall,  which  is  the  same  as  the  time 
of  moving,  is  as  y/  cb  ; therefore  the  distance  bf  is  as 

^/ac  . cb  ; and  the  distance  bc  as  y/  ad  . db.  And  hence, 
if  two  holes  are  made  equidistant  from  the  top  and  bottom, 
they  will  project  the  water  to  the  same  distance  ; for  if  ac  = 
eb,  then  the  rectangle  ac  . cb  is  equal  the  rectangle  ae  . eb  : 
which  makes  ef  the  same  for  both.  Or,  if  on  the  diameter 
ab  a semicircle  be  described  ; then,  because  the  squares  of 
the  ordinates  ch,  di,  ek  are  equal  to  the  rectangles  ac  . eb, 
&c.  ; therefore  the  distances  bf,  bg  are  as  the  ordinates 
cu,  di.  And  hence  also  it  follows,  tha?  the  projection  from 
the  middle  point  d will  be  farthest,  for  di  is  the  greatest  ordi- 
nate. 

These  are  the  proportions  of  the  distances  : but  for  the 
absolute  distances,  it  will  be  thus.  The  velocity  through 
any  hole  c,  is  such  as  will  carry  the  water  horizontally 
through  a space  equal  to  2ac  in  the  time  of  falling  through 
ac  : but,  after  quitting  the  hole,  it  describes  a parabola,  and 
comes  to  f in  the  time  a body  will  fall  through  cb  ; and 
to  find  this  distance,  since  the  times  are  as  the  roots  of 

the  spaces,  therefore  ^/ac  : y/cB  : : 2ac  : 2y/AC  . cb  = 

2ch 


PNEUMATICS. 


217 


2ch  = bf,  She  space  ranged  on  the  horizontal  plane.  And 
the  greatest  range  bg  = 2di,  or  2ao,  or  equal  to  ab. 

And  as  these  ranges  answer  very  exactly  to  the  experi- 
ments, this  confirms  the  theory,  as  to  the  velocity  assigned. 

PROPOSITION  LXXI. 

.144.  If  a Notch  or  Slit  eh  inform  of  a Parallelogram,  be  cut 
in  the  Side  of  a Vessel,  Full  of  Water,  ad  ; the  Quantity  of 
Water  flouting  through  it,  will  be  § of  the  Quantity  flowing 
through  an  equal  Orifice,  placed  at  the  Whole  Depth  eg,  or 
at  the  Base  an,  in  the  Same  Time ; it  being  supposed  that 
the  Vessel  is  always  kept  full. 

For  the  velocity  at  gh  is  to  the  velo- 
city at  il,  as  eg  to  ei  ; that  is,  as 
gh  or  il  to  ik,  the  ordinate  of  a para- 
bola ekh,  whose  axis  is  eg.  Therefore 
the  sum  of  the  velocities  at  all  the  points 
i,  is  to  as  many  times  the  velocity  at  g, 
as  the  sum  of  ail  the  ordinates  ik,  to  the 
sum  of  all  the  il’s  ; namely,  as  the  area 
of  the  parabola  egh,  is  to  the  area  eghf  ; that  is,  the 
quantity  running  through  the  noteh  eh,  is  to  the  quantity 
running  through  an  equal  horizontal  area  placed  at  gh,  as 
eghke,  to  eghf,  er  as  2 to  3 ; the  area  of  a parabola  being 
f of  its  circumscribing  parallelogram. 

345.  Corol.  1.  The  mean  velocity  of  the  water  in  the 
notch,  is  equal  to  § of  that  at  gh. 

346.  Corol.  2.  The  quantity  flowing  through  the  hole 
ighp,  is  to  that  which  would  flow  through  an  equal  orifice 
placed  as  low  as  gh,  as  the  parabolic  area  ighk,  is  to  the  rect- 
angle ighl.  As  appears  from  the  demonstration. 

OF  PNEUMATICS. 

347.  Pneumatics  is  the  science  which  treats  of  the  pro- 
perties of  air,  or  elastic  fluids. 

PROPOSITION  LXXI1. 

348.  Air  is  a Heavy  Fluid  Body  ; and  it  Surrounds  the  Earth, 
and  Gravitates  on  all  Parts  of  its  Surface. 

These  properties  of  air  are  proved  by  experience. — 
That  it  is  a fluid,  is  evident  from  its  easily  yielding  to  any 
Voe.  II.  29  4 the 


PNEUMATICS. 


218 

the  least  force  impressed  on  it,  without  making  a sensible 
resistance. 

But  when  it  is  moved  briskly,  by  any  means,  as  by  a fan 
or  a pair  of  bellows  ; or  when  any  body  is  moved  very  briskly 
through  it ; in  these  cases  we  become  sensible  of  it  as  a body, 
by  the  resistance  it  makes  in  such  motions,  and  also  by  its 
impelling  or  blowing  away  any  light  substances.  So  that, 
being  capable  of  resisting,  or  moving  other  bodies,  by  its 
impulse,  it  must  itself  be  a body,  and  be  heavy,  like  all  other 
bodies  in  proportion  to  the  matter  it  contains  ; and  therefore 
it  will  press  on  all  bodies  that  are  placed  under  it. 

A1  so,  as  it  is  a fluid,  it  spreads  itself  all  over  on  the  earth  ; 
and,  like  other  fluids,  it  gravitates  and  presses  everywhere  on 
the  earth’s  surface. 


E 

0 


= B 


1) 


349.  The  gravity  and  pressure  of  the  air 
is  also  evident  from  many  experiments. 

Thus,  for  instance,  if  water,  or  quicksilver, 
be  poured  into  the  tube  ace,  and  the  air  be 
suffered  to  press  on  it,  in  both  ends  of  the 
tube,  the  fluid  will  rest  at  the  same  height 
in  both  legs  : but  if  the  air  be  drawn  out  of 
one  end  as  e,  by  any  means  ; then  the  air 
pressing  on  the  other  end  a,  will  press  O 

down  the  fluid  in  this  leg  at  b,  and  raise  it  up  in  the  other  to 
d,  as  much  higher  than  at  b,  as  the  pressure  of  the  air  is  equal 
to.  From  which  it  appears,  not  only  that  the  air  does 
really  press,  but  also  bow  much  the  intensity  of  that 
pressure  is  equal  to.  And  this  is  the  principle  of  the 

barometer. 


PROPOSITION  LXXIII. 

350.  The  Air  is  also  an  Elastic  Fluid,  being  Condensible  and 
Expansible.  And  the  Law  it  observes  is  this,  that  its  Densi- 
ty and  Elasticity  are  proportional  to  the  Force  or  Weight 
which  Compresses  it. 

This  property  of  the  air  is  proved  by  many  experiments. 
Thus,  if  the  handle  of  a syringe  be  pushed  inward,  it  will 
condense  the  inclosed  air  into  less  space,  thereby  showing  its 
condensibility.  But  the  included  air,  thus  condensed  is 
felt  10  act  strongly  against  the  band,  resisting  the  force  com- 
pressing it  more  and  more  ; and,  on  withdrawing  the  hand, 
the  handle  is  pushed  back  again  to  where  it  was  at  first. 
Which  shows  that  the  air  is  elastic. 


351.  Again, 


ELASTICITY  OF  AIR. 


219 


351.  Again,  fill  a strong  bottle  half  fall  of 
water ; then  insert  a small  glass  tube  into 
it,  putting  its  lower  end  down  near  to  the 
bottom,  and  cementing  it  very  close  round 
the  mouth  of  the  bottle.  Then,  if  air  be 
strongly  injected  through  the  pipe,  as  by 
blowing  with  the  mouth  or  otherwise,  it 
will  pass  through  the  water  from  the  lower 
end,  ascending  into  the  parts  before  occu- 
pied with  air  at  b,  and  the  whole  mass  of 
air  become  there  condensed,  because  the 
water  is  not  compressible  into  a less  space  But,  on  remov- 
ing the  force  which  injected  the  air  a,  the  water  will  begin 
to  rise  from  thence  in  a jet,  being  pushed  up  the  pipe  by  the 
increased  elasticity  of  the  air  b,  by  which  it  presses  on  the 
surface  of  the  water,  and  forces  it  through  the  pipe,  till  as 
much  be  expelled  as  there  was  air  forced  in  ; when  the  air  at 
b will  be  reduced  to  the  same  density  as  at  first,  and  the  bal- 
ance being  restored,  the  jet  will  cease. 

352.  Likewise,  if  into  a jar  of  water 
ab,  be  inverted  an  empty  glass  tumbler 
cd,  or  such-like,  the  mouth  downward  ; 
the  water  will  enter  it,  and  partly  fill  it, 
but  not  near  so  high  as  the  water  in  the 
jar,  compressing  and  condensing  the  air 
into  a less  space  in  the  upper  parts  c,  and 
causing  the  glass  to  make  a sensible  resis- 
tance to  the  hand  in  pushing  it  down. 

Then,  on  removing  the  hand,  the  elasticity  of  the  internal 
condensed  air  throws  the  glass  up  again.  All  these  showing 
that  the  air  is  condensible  and  elastic. 


353.  Again,  to  show  the  rate  or  proportion 
of  the  elasticity  to  the  condensation  : take  a 
long  crooked  glass  tube,  equally  wade  through- 
out, or  at  least  in  the  part  bd,  and  open  at  a, 
but  close  at  the  other  end  b.  Pour  in  a little 
quicksilver  at  a,  just  to  cover  the  bottom  to 
the  bend  at  cd,  and  to  stop  the  communica- 
tion between  the  external  air  and  the  air  in 
bd.  Then  pour  in  more  quicksilver,  and 
mark  the  corresponding  heights  at  which  it 
stands  in  the  two  legs  : so,  when  it  rises  to 
h in  the  open  leg  ac,  let  it  rise  to  e in  the 
close  one,  reducing  its  included  air  from  the 
natural  bulk  bd  to  the  contracted  space  be. 


220 


PNEUMATICS. 


by  the  pressure  of  the  column  He  ; and  when  the  quick- 
silver stands  at  i and  k,  in  the  open  leg,  let  it  rise  to  f and  r, 
in  the  other,  reducing  the  air  to  the  respective  spaces  bf, 
bg,  by  the  weights  of  the  columns  if,  Kg.  Then  it  is  al- 
ways found,  that  the  condensations,  and  elasticities  are  as 
the  compressing  weights  or  columns  of  the  quicksilver, 
and  the  atmosphere  together.  So,  if  the  natural  bulk  of 
the  air  bd  he  compressed  into  the  spaces  be,  bf,  bg,  which 
are  f , f , a of  bd,  or  as  the.  numbers  3,  2,  1 : then  the  at- 
mosphere, together  with  the  corresponding  columns  He,  if, 
Kg,  are  also  found  to  be  in  the  same  proportion  reciprocally, 
viz,  as  J , i,  or  as  the  numbers  2,  3,  6.  And  then  He  = 
iA,  if  — a,  and  Kg  = 3a  ; where  a is  the  weight  of  atmos- 
phere Which  show,  that  the  condensations  are  directly 
as  the  compressing  forces-  And  the  elasticities  are  in  the 
same  ratio,  since  the  columns  in  ac  are  sustained  by  the  elasti- 
cities in  bd. 

From  the  foregoing  principles  may  be  deduced  many  useful 
remarks,  as  in  the  following  corollaries,  viz. 

334.  Cnrol.  1.  The  space  which 
any  quantity  of  air  is  confined  in, 
is  reciprocally  as  the  force  that 
compresses  it.  So,  the  forces  which 
confine  a quantity  of  air  in  the  cy- 
lindrical .spaces  ag,  bg,  cc,  are 
reciprocally  as  the  same,  or  reci- 
procally as. the  heights  ad,  bd,  cd. 

And  therefore  if  to  the  two  per- 
pendicular lines  da,  dh,  as  asymptotes,  the  hyperbola  ikl 
be  described,  and  the  ordinates  ai,  bk,  cl  be  drawn  ; then 
the  forces  which  confine  the  air  in  the  spaces  ag,  eg,  cg, 
will  be  directly  as  the  corresponding  ordinates  ai,  bk,  cl, 
since  these  are  reciprocally  as  the  abscisses  ad,  bd,  cd, 

by  the  nature  of  tiae  hyperbola. 

355.  Corol.  2.  All  the  air  near  the  earth,  is  in  a state  of 
compression,  by  the  weight  of  the  incumbent  atmosphere. 

336  Corol.  3.  The  air  is  denser  near  the  earth,  than  in 
high  places  ; or  denser  at  the  foot  of  a mountain,  than  at 
the  top  of  it.  And  the  higher  above  the  earth,  the  less  dense 
it  is. 

357.  Corol.  4.  The  spring  or  elasticity  of  the  air,  is  equal 
to  the  weight  of  the  atmosphere  above  it  ; and  they  will  pro- 
duce the  same  effects  : since  they  always  sustain  and  balance 
each  other. 

358.  Corol.  5. 


ELASTICITY  OF  AIR. 


221 


358.  Corol.  5.  If  the  density  of  the  air  be  increased,  pre- 
serving the  same  heat  or  temperature  its  spring  or  elasticity  is 
also  increased,  and  in  the  same  proportion. 

359j  Corol.  6.  By  the  pressure  and  gravity  of  the  atmos- 
phere, on  the  surface  of  the  fluids,  the  fluids  are  made  to  rise  in 
any  pipes  or  vessels,  when  the  spring  or  pressure  within  is 
decreased  or  taken  off". 

PROPOSITION  LXXIV. 

360.  Heat  Increases  the  Elasticity  of  the  Air , and  Cold  Dimin- 
ishes it.  Or,  Heat  Expands,  and  Cold  Condenses  the  Air. 

This  property  is  also  proved  by  experience. 

361.  Thus,  tie  a bladder  very  close  with  some  air  in  it  ; 
and  lay  it  before  the  fire.  : then  as  it  warms,  it  will  more  and 
more  distend  the  bladder,  and  at  last  burst  it,  if  the  heat  be 
continued  and  increased  high  enough.  But  if  the  bladder 
be  removed  from  the  fire,  as  it  cools  it  will  contract  again, 
as  before.  And  it  was  on  this  principle  that  the  first  air- 
balloons  were  made  by  Montgolfier  : for,  by  heating  the  air 
within  them,  by  a fire  beneath,  the  hot  air  distends  them  to  a 
size  which  occupies  a space  in  the  atmosphere,  whose  weight  of 
common  air  exceeds  that  of  the  balloon. 

362.  Also,  if  a cup  or  glass,  with  a little  air  in  it,  be  invert- 
ed into  a vessel  of  water  ; and  the  whole  be  heated  over 
the  fire  or  otherwise  ; the  air  in  the  top  will  expand  til!  it  fill 
the  glass,  and  expel  the  water  out  of  it  ; and  part  of  the  air 
itself  will  follow,  by  continuing  or  increasing  the  heat. 

Many  other  experiments,  to  the  same  effect,  might  be 
adduced,  all  proving  the  properties  mentioned  in  the  propo  - 
sition. 

SCHOLIUM. 

363.  So  that,  when  the  force  of  the  elasticity  of  air  is  con- 
sidered, regard  must  be  had  to  its  heat  ©r  temperature  ; the 
same  quantity  of  air  being  more  or  less  elastic,  as  its  heat  is 
more  or  less.  And  it  has  been  found,  by  experiment,  that  the 
elasticity  is  increased  by  the  435th  part,  for  each  degree  of 
heat,  of  which  there  are  180  between  the  freezing  and  boiling 
heat,  of  water. 

364.  N.  B.  Water  expands  about  the  g oo  part,  with  each 

degree  of  heat.  (Sir  Geo.  Shuckburgh,  Bhilos.  Trans.  1777, 
f.  560,  &c.)  Also, 


222 


PNEUMATICS. 


Also,  tbe 

Spec.  grav.  of  air  1*201  or  If  i 
water  1000  > 
mercury  13592  ) 

Or  thus,  air  1-222  or  If  ) 
water  1000  > 
mercury  13600) 


when  the  barom.  is  29-5, 
and  the  therm,  is  55° 
which  are  their  mean  heights 
in  this  country. 

when  the  barom.  is  30, 
and  thermometer  55. 


PROPOSITION  LXXV. 


365.  The  Weight  or  Pressure  of  the  Atmosphere,  on  any  Base  at  I 
the  Earth’s  Surface,  is  Equal  to  the  Weight  of  a Column  of  1 
Quicksilver,  of  the  Same  Base,  and  the  Height  of  which  is  be-  j 
tween  28  and  31  inches. 

This  is  proved  by  the  barometer,  an  instrument  which 
measures  the  pressure  of  the  air,  and  which  is  described  i 
below.  For,  at  some  seasons,  and  in  some  places,  the  air 
sustains  and  balances  a column  of  mercury,  of  about  28 
inches  : but  at  other  times  it  balances  a column  of  29,  or  30, 
or  near  31  inches  high  ; seldom  in  the  extremes  28  or  31, 
but  commonly  about  the  means  29  or  30.  A variation  w-hich 
depends  partly  on  the  different  degrees  of  heat  in  the  air  near 
the  surface  of  the  earth,  and  partly  on  the  commotions  and 
changes  in  the  atmosphere,  from  winds  and  other  causes,  by 
which  it  is  accumulated  in  some  places,  and  depressed  in 
others,  being  thereby  rendered  denser  and  heavier,  or  rarer 
and  lighter  ; which  changes  in  its  state  are  almost  continually 
happening  in  any  one  place.  But  the  medium  state  is  com- 
monly about  29f  or  30  inches. 


366.  Corol.  1.  Hence  the  pressure  of  the  atmosphere  on 
every  square  inch  at  the  earth’s  surface,  at  a medium,  is  very 
near  15  pounds  avoirdupois,  or  rather  14f  pounds.  For,  a 
cubic  foot  of  mercury  weighing  13600  ounces  nearly,  an 
inch  of  it  will  weigh  7-866  or  almost  8 ounces,  or  nearly 
half  a pound,  which  is  the  weight  of  the  atmosphere  for 
every  inch  of  the  barometer  on  a base  of  a square  inch  ; and 
therefore  30  inches,  or  the  medium  height,  weighs  very  near 
14£  pounds. 

367.  Corol.  2.  Hence  also  the  weight  or  pressure  of  the 
atmosphere,  is  equal  to  that  of  a column  of  water  from  32 
to  35  feet  high,  or  on  a medium  33  or  34  feet  high.  For, 
water  and  quicksilver  are  in  weight  nearly  as  1 to  13-6  ; 

so 


PRESSURE  OF  THE  ATMOSPHERE. 


223 


so  that  the  atmosphere  will  balance  a column  of  water  13-6 
times  as  high  as  one  of  quicksilver  ; consequently 

13-6  times  28  inches  = 381  inches,  or  31 1 feet, 

13-6  times  25  inches  = 394  inches,  or  32f  feet, 

13-6  times  30  inches  = 408  inches,  or  34  feet, 

13 -6  times  31  inches  = 422  inches,  or  35}  feet. 

And  hence  a common  sucking  pump  will  not  raise  water 
higher  than  about  33  or  34  feet.  And  a siphon  will  not  run, 
if  the  perpendicular  height  of  the  top  of  it  be  more  than 
about  33  or  34  feet. 


368.  Corol.  3.  If  the  air  were  of  the  same  uniform  den- 
sity at  every  height  up  to  the  top  of  atmosphere,  as  at 
the  surface  of  the  earth  ; its  height  would  be  about  5i 
miles  at  a medium.  For,  the  weights  of  the  same  bulk  of 
air  and  water,  are  nearly  as  T222  to  1000  ; therefore  as 
1-222  : 1000  : : 3Sf  feet  : 27600  feet,  or  5}  miles  nearly. 
And  so  high  the  atmosphere  would  be,  if  it  were  all  of 
uniform  density,  like  water.  But,  instead  of  that,  from 
its  expansive  and  elastic  quality,  it  becomes  continually 
more  and  more  rare,  the  farther  above  the  earth,  in  a cer- 
tain proportion,  which  will  be  treated  of  below,  a3  also  the 
method  of  measuring  heights  by  the  barometer,  which  de- 
pends on  it. 


369.  Corol.  4.  From  this  proposition  and  the  last  it  fol- 
lows, that  the  height  is  always  the  same,  of  an  uniform 
atmosphere  above  any  place,  which  shall  be  all  of  the  uni- 
form density  with  the  air  there,  and  of  equal  weight  or 
pressure  with  the  real  height  of  the  atmosphere  above  that 
place,  whether  it  be  at  the  same  place,  at  different  times, 
or  at  any  different  places  or  heights  above  the  earth  ; and 
that  height  is  always  about  miles,  or  27600  feet,  as 
above  found.  For,  as  the  density  varies  in  exact  propor- 
tion to  the  weight  of  the  column,  therefore  it  requires  a 
column  of  the  same  height  in  all  cases,  to  make  the  re- 
spective weights  or  pressures.  Thus,  if  w and  zv  be  the 
weights  of  atmosphere  above  any  places,  d and  cl  their 
densities,  and  h and  h the  heights  of  the  uniform  columns, 
©f  the  same  densities  and  weights  ; Then  h X d = w,  and 


h X d = w ; therefore  — or  h is  equal  to  — or  h. 

d 1 d 


The  tem- 


perature being  the  same. 


PROPOSITION 


224 


PNEUMATICS. 


PROPOSITION  LXXVI. 

370.  The  Density  of  the  Atmospnere,  at  Different  Heights  above 
the  Earth,  Decreases  in  such  Sort,  that  when  the  Heights  In- 
crease in  Arithmetical  Progression,  the  Densities  Decrease  in 
Geometrical  Progression. 


Let  the  indefinite  perpendicular  line  ap, 
erected  on  the  earth,  be  conceived  to  be  divided 
into  a great  number  of  very  small  equal  parts, 
a,  b,  c,  n,  &c.  forming  so  many  thin  strata  of 
air  in  the  atmosphere,  all  of  different  density, 
gradually  decreasing  from  the  greatest  at  a : 
then  the  density  of  the  several  strata,  a,  b,  c, 
d,  &c.  will  be  in  geometrical  progression  de- 
creasing. 

For,  as  the  strata  a,  b,  c,  &c.  are  all  of  equal 
thickness,  the  quantity  of  matter  in  each  of  them,  is  as  the 
density  there  ; but  the  density  in  any  one,  being  as  the  com- 
pressing  force,  is  as  the  weight  or  quantity  of  all  the  matter 
from  that  place  upward  to  the  top  of  the  atmosphere  ; there-  i 
fore  the  quantity  of  matter  in  each  stratum,  is  also  as  the 
whole  quantity  from  that  place  upward.  Now,  if  from  the 
whole  weight  at  any  place  as  b,  the  weight  or  quantitj7  in  the 
stratum  b be  subtracted,  the  remainder  is  the  weight  at  the 
next  stratum  c ; that  is,  from  each  weight  subtracting  a part 
which  is  proportional  to  itself,  leaves  the  next  weight  ; or, 
which  is  the  same  thing,  from  each  density  subtracting  a 
part  which  is  proportional  to  itself,  leaves  the  next  den- 
sity. But  when  any  quantities  are  continually  diminished  by 
parts  which  are  proportional  to  themselves,  the  remainders 
form  a series  of  continued  proportionals  : consequently  these 
densities  are  in  geometrical  progression. 

Thus,  if  the  first  density  be  d,  and  from  each  be  taken 

its  nth  part : there  will  then  remain  its  - — part,  or  the  - 

r n n 


part  putting  m for  n — 1 ; and  therefore  the  series  of  den- 
sities will  be  n,  - d,  — d,  — d,  — d,  &c.  the  common  ratio 

71  7l3  n* 


of  the  series  being  that  of  n to  m. 


SCHOLIUM. 


371.  Because  the  terms  of  an  arithmetical  series,  are  pro- 
portional to  the  logarithms  of  the  terms  of  a geometrical 
series  : therefore  different  altitudes  above  the  earth's  sur- 
face, 


DENSITY  OF  THE  ATMOSPHERE. 


225 


' face,  are  as  the  logarithms  of  the  densities,  or  of  the  weights 
of  air,  at  those  altitudes. 

So  that,  if  d denote  the  density  at  the  altitude  a, 
and  d - the  density  at  the  altitude  a ; 
then  a being  as  the  log.  of  d,  and  a as  the  log.  of  d, 

the  dif.  of  alt.  a — a will  be  as  the  log.  d — log.  d.  or  log. 

And  if  a = 0,  or  d the  density  at  the  surface  of  the  earth  5 
then  any  altitude  above  the  surface  a,  is  as  the  log.  of 

Or,  in  general,  the  log.  of  ^ is  as  the  altitude  of  the  one 

place  above  the  other,  whether  the  lower  place  be  at  the 
surface  of  the  earth,  or  any  where  else. 

And  from  this  property  is  derived  the  method  of  deter- 
mining the  heights  of  mountains  and  other  eminences,  by 
the  barometer,  which  is  an  instrument  that  measures  the 
pressure  or  density  of  the  air  at  any  place.  For,  by  taking, 
with  this  instrument,  the  pressure  or  density,  at  the  foot  of 
a hill  for  instance,  and  again  at  the  top  of  it,  the  differ- 
ence of  the  logarithms  of  these  two  pressures,  or  the  loga- 
rithm of  their  quotient,  will  be  as  the  difference  of  altitude, 
or  as  the  height  of  the  hill  ; supposing  the  temperatures  of 
the  air  to  be  the  same  at  both  places,  and  the  gravity  of  air 
not  altered  by  the  different  distances  from  the  earth’s 
l centre. 

372.  But  as  this  formula  expresses  only  the  relations  be- 
tween different  altitudes  with  respect  to  their  densities,  re- 
course must  be  had  to  some  experiment,  to  obtain  the  real 
altitude  which  corresponds  to  any  given  density,  or  the  den- 
sity which  corresponds  to  a given  altitude.  And  there  are 
various  experiments  by  which  this  may  be  done.  The  first, 
and  most  natural,  is  that  which  results  from  the  known  spe- 
Jcific  gravity  of  air,  with  respect  to  the  whole  pressure  of  the 
atmosphere  on  the  surface  of  the  earth.  Now,  as  the  alti- 
tude a is  always  as  log.  § ; assume  h so  that  a = h X log. 

d d 

where  h will  be  of  one  constant  value  for  all  altitudes  ; and  to 
determine  that  value,  let  a case  be  taken  in  which  we  know 
the  altitude  a corresponding  to  a known  density  d ; as  for 
instance,  take  a = 1 foot,  or  1 inch,  or  some  such  small  al- 
titude ; then,  because  the  density  d may  be  measured  by  the 
pressure  of  the  atmosphere,  or  the  uniform  column  of  27600 
feet,  when  the  temperature  is  55°  ; therefore  27600  feet  will 

Vol.  II.  30  denote 


226 


PNEUMATICS. 


denote  the  density  d at  the  lower  place,  and  27599  the  less 


density  d at  1 foot  abbve  it  ; consequently  1 = h X log. 


27600 

27599’ 


which,  by  the  nature  of  logarithms,  is  nearly 


, -43429448 

27600 


= ^7—^  nearly  ; and  hence  h = 63551  feet  ; which  gives, 
for  any  altitude  in  general,  this  theorem,  viz.  a — 63551  X 
log.  or  = 63551  X log.  - feet,  or  10592  X log.  - 

d m m 

fathoms  ; where  m is  the  column  of  mercury  which  is  equal 
to  the  pressure  or  weight  of  the  atmosphere  at  the  bottom, 
and  rn  that  at  the  top  of  the  altitude  a ; and  where  m and  m 
may  be  taken  in  any  measure,  either  feet  or  inches,  &c. 


373.  Note,  that  this  formula  is  adapted  to  the  mean  tem-  j 
perature  of  the  air  55°.  But,  for  every  degree  of  tempe- 
rature ditfereiit  from  this,  in  the  medium  between  the  tem- 
peratures at  the  top  and  bottom  of  the  altitude  a,  that  alti- 
tude will  vary  by  its  435th  part  ; which  must  be  added,  when 
that  medium  exceeds  55°,  otherwise  subtracted. 

I 

374.  Note,  also,  that  a column  of  30  inches  of  mercury  ; 
varies  its  length  by  about  the  part  of  an  inch  for  ever}' 
degree  of  heat;  or  rather  g-g-V o °f  the  whole  volume. 

375.  But  the  formula  may  be  rendered  much  more  con- 
venient for  use,  by  reducing  the  factor  10592  to  10000,  by 
changing  the  temperature  proportionally  from  55°  ; thus, 
as  the  diff.  592  is  the  18th  part  of  the  whole  factor  10592  ; 
and  as  18  is  the  24th  part  of  435  ; therefore  the  correspond-  i 
ing  change  of  temperature  is  2 i°,  which  reduces  the  55°  to 

31°.  So  that  the  formula  is,  a — 10000  X log.—  fathoms, 

VI 

when  the  temperature  is  31  degrees  ; and  for  every  degree 
above  that,  the  result  is  to  be  increased  by  so  many  times  its 
435th  part. 

378.  Exam.  1.  To  find  the  height  of  a hill  when  the 
piessure  of  the  atmosphere  is  equal  to  29-68  inches  of  mer- 
cury at  the  bottom,  and  25  28  at  the  top  ; the  mean  tem- 
perature being  50®  ? Ans.  4378  feet,  or  730  fathoms. 


377.  Exam.  2.  To  find  the  height  of  a hill  when  the 
atmosphere  weighs  29-45  inches  of  mercury  at  the  bottom, 
and  26-82  at  the  top,  the  mean  temperature  being  33®  7 

Ans.  2386  feet,  or  397a  fathoms. 

378.  Exam.  3. 


THE  SIPHON. 


227 


378.  Exam.  3.  At  what  altitude  is  the  density  of  the  at- 
mosphere only  the  4th  part  of  what  it  is  at  the  earth’s  sur- 
face ? Ans  6020  fathoms. 

By  the  weight  and  pressure  of  the  atmosphere,  the  effect 
and  operations  of  pneumatic  engines  may  be  accounted  for, 
and  explained  ; such  as  siphons,  pumps,  barometers,  &c.  ; of 
which  it  may  not  be  improper  here  to  give  a brief  description. 


OF  THE  SIPHON. 

379.  The  Siphon,  or  Syphon,  is  any 
bent  tube,  having  its  two  legs  either 
of  equal  or  of  unequal  length. 

If  it  be  filled  with  water,  and  then 
inverted,  with  the  two  open  ends 
downward,  and  held  level  in  that  posi- 
tion ; the  water  will  remain  suspended 
in  it,  if  the  two  legs  be  equal  For  the 
j atmosphere  will  press  equally  on  the 
, surface  of  the  water  in  each  end,  and 
support  them,  if  they  are  not  more  than  34  feet  high  ; and  the 
legs  being  equal,  the  water  in  them  is  an  exact  counterpoise 
by  their  equal  weights  ; so  that  the  one  has  now  power  to 
move  more  than  the  other  ; and  they  are  both  supported  by 
the  atmosphere. 

But  if  now  the  siphon  be  a little  inclined  to  one  side,  sg 
that  the  orifice  of  one  end  be  lower  than  that  of  the  other  ; 
or  if  the  legs  be  of  unequal  length,  which  is  the  same  thing  ; 
then  the  equilibrium  is  destroyed,  and  the  water  will  all  de- 
scend out  by  the  lower  end,  and  rise  up  in  the  higher. 
For,  the  air  pressing  equally,  but  the  two  ends  weighing 
unequally,  a motion  must  commence  where  the  power  is 
greatest,  and  so  continue  till  all  the  water  has  run  out  by  the 
lower  end.  And  if  the  shorter  leg  be  immersed  into  a vessel 
of  water,  and  the  siphon  be  set  a running  as  above,,  it  will 
continue  to  run  till  all  the  water  be  exhausted  out  of  the 
vessel,  or  at  least  as  low  as  that  end  of  the  siphon.  Or,  it 
may  be  set  a running  without  filling  the  siphon  as  above,  by 
only  inverting  it,  with  its  shorter  leg  into  the  vessel  of  water  ; 
then,  with  the  mouth  applied  to  the  lower  orifice  a,  suck 
the  air  out ; and  the  water  will  presently  follow,  being  forced 
up  in  the  siphon  by  the  pressure  of  the  air  on  the  water 
in  the  vessel. 


OF 


[ 228  j 

OF  THE  PUMP. 


380.  There  are  three  sorts 
of  pumps  : the  Sucking,  the 
Lifting,  and  the  Forcing  Pump. 

By  the  first,  water  can  be  raised 
only  to  about  34  feet,  viz.  by 
the  pressure  of  the  atmosphere  ; 
but  bv  the  others,  to  any  height  ; 
but  then  they  require  more  ap- 
paratus and  power 
The  annexed  figure  represents 
a common  sucking  pump,  ab 
is  the  barrel  of  the  pump,  being 
a hollow  cylinder,  made  of  me- 
tal, and  smooth  within,  or  of 
wood  for  very  common  pur- 
poses. cd  is  the  handle,  move- 
able  about  the  pin  e,  by  moving 
the  end  c up  and  down,  df 
an  iron  rod  turning  about  a pin 
d,  which  connects  it  to  the 
end  of  the  handle.  This  rod  is  fixed  to  the  piston,  bucket, 
or  sucker,  fg,  by  which  this  is  moved  up  and  down  within 
the  barrel,  which  it  must  fit  very  tight  and  close,  that  no  air 
or  water  may  pass  between  the  piston  and  the  sides  of  the 
barrel ; and  for  this  purpose  it  is  commonly  armed  with 
leather.  The 'piston  is  made  hollow,  or  it  has  a perforation 
through  it,  the  orifice  of  which  is  covered  by  a valve  h 

opening  upwards,  i is  a’plug  firmly  fixed  in  the  lower  part 

of  the  barrel,  also  perforated,  and  covered  by  a valve  k open- 
ing upwards. 

381.  When  the  pump  is  first  to  be  worked,  and  the  water 
is  below  the  plug  i ; raise  the  end  c of  the  handle,  then  the 
piston  descending,  compresses  the  air  in  hi,  which  by  its 
spring  shuts  fast  the  valve  k,  and  pushes  up  the  valve  h; 
and  so  enters  into  the  barrel  above  the  piston.  Then  put- 
ting the  end  c of  the  handle  down  again,  raises  the  piston 

or  sucker,  which  lifts  up  with  it  the  column  of  air  above  it, 
the  external  atmosphere  by  its  pressure  keeping  the  valve  h 
shut : the  air  in  the  barrel  being  thus  exhausted,  or  rarefied, 
is  no  longer  a counterpoise  to  that  which  presses  on  the  sur- 
face of  the  water  in  the  well  ; this  is  forced  up  the  pipe,  and 
through  the  valve  k,  into  the  barrel  of  the  pump.  Then 
pushing  the  piston  down  again  into  this  water,  now  in  the 

barrel. 


THE  AIR-PUMP. 


229 


barrel  its  weight  shuts  the  lower  valve  k,  and  its  resistance 
forces  up  the  valve  of  the  piston,  and  enters  the  upper  part 
of  the  barrel,  above  the  piston.  Then,  the  bucket  being 
raised,  lifts  up  with  it  the  water  which  had  passed  above  its 
valve,  and  it  runs  out  by  the  cock  l ; and  taking  off  the 
weight  below  it,  the  pressure  of  the  external  atmosphere  on 
the  water  in  the  well  again  forces  it  up  through  the  pipe  and 
lowe[  valve  close  to  the  piston,  all  the  way  as  it  ascends,  thus 
keeping  the  barrel  alwaj's  full  of  water.  And  thus  by  re- 
peating the  strokes  of  the  piston,  a continued  discharge  is 
made  at  the  cock  l. 


OF  THE  AIR-PUMP. 

382.  Nearly  on  the  same  principles  as  the  water  pump, 
is  the  invention  of  the  Air-pump,  by  which  the  air  is  drawn 
out  of  any  vessel,  like  as  water  is  drawn  out  by  the  former. 
A brass  barrel  is  bored  and  polished  truly  cylindrical,  and  ex- 
actly fitted  with  a turned  piston,  so  that  no  air  can  pass  by 
the  sides  of  it,  and  furnished  with  a proper  valve  opening 
upward.  Then  by  lifting  up  the  piston,  the  air  in  the  close 
vessel  below  it  follows  the  piston  and  fills  the  barrel  ; and 
being  thus  diffused  through  a larger  space  than  before,  when 
it  occupied  the  vessel  or  receiver  only,  but  not  the  barrel, 
it  is  made  rarer  than  it  was  before,  in  proportion  as  the  ca- 
pacity of  the  barrel  and  receiver  together  exceeds  the  re- 
ceiver alone.  Another  stroke  of  the  piston  exhausts  another 
barrel  of  this  now  rarer  air,  which,  again  rarefies  it  in  the 
same  proportion  as  before.  And  so  on,  for  any  number  of 
strokes  of  the  piston,  still  exhausting  in  the  same  geometri- 
cal progression,  of  which  the  ratio  is  that  which  the  capacity 
of  the  receiver  and  barrel  together  exceeds  the  receiver,  till 
this  is  exhausted  to  any  proposed  degree,  or  as  far  as  the  na- 
ture of  the  machine  is  capable  of  performing;  which  happens 
when  the  elasticity  of  the  included  air  is  so  far  diminished,  by 
rarefying,  that  it  is  too  feeble  to  push  up  the  valve  of  the  pis- 
ton and  escape. 

383.  From  the  nature  of  this  exhausting,  in  geometrical 
progression,  we  may  easily  find  how  much  the  air  in  the  re- 
ceiver is  rarefied  by  any  number  of  strokes  of  the  piston  ; or 
what  number  of  such  strokes  is  necessary,  to  exhaust  the  re- 
ceiver to  any  given  degree.  Thus,  if  the  capacity  of  the  re- 
ceiver and  barrel  together,  be  to  that  of  the  receiver  alone. 

• , as 


230 


OF  THE  DIVING  BELL. 


as  c to  r,  and  1 denote  the  natural  density  of  the  air  at  first  : 
then 

c : r : : 1 : — , the  density  after  one  stroke  of  the  piston. 

V 7*2 

c : r : : - : — , the  density  after  2 strokes, 
r2  r 3 

c : r : : — : — , the  density  after  three  strokes, 

rn 

&c.  and  ~r  tne  density  after  n strokes. 

cn 

So,  if  the  barrel  be  equal  to  ^ of  the  receiver  ; then  c : r : . 
4n 

5:4;  and  — = 0-8n  is  = d the  density  after  n turns.  And 

if  n be  20,  then  0-82  0 = '0115  is  the  density  of  the  included 
air  after  20  strokes  of  the  piston  ; which  being  the  86^  part 
of  1»  or  the  first  density,  it  follows  that  the  air  is  86T7^  times 
rarefied  by  the  20  strokes. 

384.  Or,  if  it  were  required  to  find  the  number  of  strokes 
necessary  to  rarefy  the  air  any  number  of  times  ; because 

— is  = the  proposed  density  d ; therefore,  taking  the  loga- 

Ln 

T Joe*,  cl 

rithms,  n X log.  — = log.  d,  and  n = , — — , thenurn- 
c 1.  r — 1.  c 

her  of  strokes  required.  So  if  r be  f of  c,  and  it  he  re- 
quired to  rarify  the  air  100  times  : then  d = T£¥  or  0-1  ; 

and  hence  n = = 20f  nearly.  So  that  in  205 

strokes  the  air  will  be  rarefied  100  times. 


OF  THE  DIVING  BELL  & CONDENSING  MACHINE. 

385.  On  the  same  principles  too  depend  the  operations 
and  effect  of  the  Condensing  Engine,  by  which  air  may  be 
condensed  to  any  degree  instead  of  rarefied  as  in  the  air- 
pump.  And,  like  as  the  air-pump  rarefies  the  air,  by  ex- 
tracting always  one  barrel  of  air  after  another  : so,  by  this 
other  machine,  the  air  is  condensed,  by  throwing  in  or  add- 
ing always  one  barrel  of  air  after  another  ; which  it  is 
evident  may  be  done  by  only  turning  the  valves  of  the 
piston  and  barrel,  that  is,  making  them  to  open  the  con- 
trary way,  and  working  the  piston  in  the  same  manner  ; 

so 


OF  THE  DIVING  BELL. 


231 


so  that,  as  they  both  open  upward  or  outward  in  the  aL-pump 
or  rarefier,  they  will  both  open  downward  or  inward  in  the 
condenser. 


386.  And  on  the  same  principles,  namely,  of  the  com- 
pression and  elasticity  of  the  air,  depends  the  use  of  the  Div- 
ing Bell,  which  is  a large  vessel,  in  which  a person  descends 
to  the  bottom  of  the  sea,  the  open  end  of  the  vessel  being 
downward  ; only  in  this  case  the  air  is  not  condensed  by  forc- 
ing more  of  it  into  the  same  space,  as  in  the  condensing  en- 
gine ; but  by  compressing  the  same  quantity  of  air  into  a less 
space  in  the  bell,  by  increasing  always  the  force  which  com- 
presses it. 


387.  If  a vessel  of  any  sort  be  inverted  into  water,  and 
pushed  or  let  down  to  any  depth  in  it ; then  by  the  pressure 
of  the  water  some  of  it  will  ascend  into  the  vessel,  but  not  so 
high  as  the  water  without,  and  will  compress  the  air  into  less 
space,  according  to  the  difference  between  the  heights  of  the 
internal  and  external  water  ; and  the  density  and  elastic  force 
of  the  air  will  be  increased  in  the  same  proportion,  as  its  space 
in  the  vessel  is  diminished. 

So,  if  the  tube  ce  be  inverted,  and  pushed  down  into 
water,  till  the  external  water  exceed  the  internal,  by  the 
height  ab,  and  the  air  of  the  tube  be  reduced  to  the  space 
cd  ; then  that  air  is  pressed  both  by  a co- 
lumn of  water  of  the  height  ab,  and  by  the 
whole  atmosphere,  which  presses  on  the 
upper  surface  of  the  water  ; consequently 
the  space  cd  is  to  the  whole  space  ce,  as 
the  weight  of  the  atmosphere,  is  to  the 
weights  both  of  the  atmosphere  and  the 
column  of  water  ab.  So  that  if  ab  be 
about  34  feet,  which  is  equal  to  the  force 
of  the  atmosphere,  then  cd  will  be  equal 
to  ^ce  ; but  if  ab  be  double  of  that,  or  68 
feet,  then  cd  will  be  Ice  ; and  so  on.  And  hence,  by  know- 
ing the  depth  af,  to  which  the  vessel  is  sunk,  we  can  easily 
find  the  point  d,  to  which  the  water  will  rise  within  it  at  any 
time.  For  let  the  weight  of  the  atmosphere  at  that  time  be 
equal  to  that  of  34  feet  of  water  ; also,  let  the  depth  af  be  20 
feet,  and  the  length  of  the  tube  ce  4 feet  : then  putting  the 
height  of  the  internal  water  de  = x , 
it  is  34  + ab  : 34  : : ce  : cd, 
that  is  34  + af  — de  : 34  : : ce  : ce  — de, 
or  54  — x : 34  : : 4 : 4 — x ; 
hence,  multiplying  extremes  and  means,  216  — 53„r  4-  x2  = 

136, 


232 


THE  BAROMETER. 


13G,  and  the  root  is  x = 2 very  nearly  = 1 -4 14  of  a foot, 

or  17  inches  nearly  ; being  the  height  de  to  which  the  water 
will  rise  within  the  tube. 


388.  But  if  the  vessel  be  not  equally 
wide  throughout,  but  of  any  other 
shape,  as  of  a bell-like  form,  such  as 
is  used  in  diving  ; then  the  altitudes 
will  not  observe  the  proportion  above, 
but  the  spaces  or  bulks  only  will  re- 
spect that  proportion,  namely,  34  + 
ab  : 34  : : capacity  ckl  : capacity 
chi,  if  it  be  common  or  fresh  water  ; 
and  33  + ab  : 33  : : capacity  ckl  : 
capacity  chi,  if  it  be  sea-water.  From 
which  proportion,  the  height  de  may 
be  found,  when  the  nature  or  shape  of 
is  known. 


the  vessel  or  bell  ckl 


OF  THE  BAROMETER. 

389.  THE  Barometer  is  an  instrument  for  measuring  the 
pressure  of  the  atmosphere,  and  elasticity  of  the  air,  at  any 
time.  It  is  commonly  made  of  a glass  tube,  of  near  3 feet 
long,  close  at  one  end,  and  tilled  with  mercury.  When  the 
tube  is  full,  by  stopping  the  open  end  with  the  finger,  then 
inverting  the  tube,  and  immersing  that  end  with  the  finger  into 
a bason  of  quicksilver,  on  removing  the  finger  from  the  ori- 
fice, the  fluid  in  the  tube  will  descend  into  the  bason,  till  what 
remains  in  the  tube  be  of  the  same  weight  with  a column  of 
the  atmosphere,  which  is  commonly  between  28  and  31  inches 
of  quicksilver  ; and  leaving  an  entire  vacuum  in  the  upper 
end  of  the  tube  above  the  mercury.  For,  as  the  upper  end 
of  the  tube  is  quite  void  of  air,  there  is  no  pressure  down- 
wards but  from  the  column  of  quicksilver,  and  therefore  that 
will  be  an  exact  balance  to  the  counter  pressure  of  the  whole 
column  of  atmosphere,  acting  on  the  orifice  of  the  tube  by  the 
quicksilver  in  the  bason.  The  upper  3 inches  of  the  tube, 
namely,  from  28  to  31  inches,  have  a scale  attached  to  them, 
divided  into  inches,  tenths,  and  hundredths,  for  measuring  the 
length  of  the  column  at  all  times,  by -observing  which  division 
of  the  scale  the  top  of  the  quicksilver  is  opposite  to  ; as  it  as- 
cends and  descends  within  these  limits  according  to  the  state 
of  the  atmosphere. 


So 


THE  THERMOMETER. 


233 


So  that  the  weight  of  the  quick- 
silver in  the  tube,  above  that  in 
the  bason,  is  at  all  times  equal  to 
the  weight  or  pressure  of  the  co- 
lumn of  atmosphere  above  it,  and 
of  the  same  base  with  the  tube  ; 
and  hence  the  weight  of  it  may 
at  all  times  be  computed ; being 
nearly  at  the  rate  of  half  a pound 
avoirdupois  for  every  inch  of  quick- 
silver in  the  tube,  on  every  square 
inch  of  base  ; or  more  exactly  it 
is  of  a pound  on  the  square 
inch,  for  every  inch  in  the  altitude 
of  the  quicksilver  weighs  just  T5¥9¥lb, 
or  nearly  i a pound,  in  the  mean 
temperature  of  55°  of  heat.  And 
consequently,  when  the  barometer 
stands  at  30  inches,  or  feet  high, 
which  is  nearly  the  medium  or 
standard  height,  the  whole  pressure 
of  the  atmosphere  is  equal  to  14f 
pounds  on  every  square  inch  of  the  base  ; and  so  in  propor- 
tion for  other  heights. 


OF  THE  THERMOMETER. 

■ 390.  THE  Thermometer  is  an  instrument  for  measuring 

the  temperature  of  the  air,  as  to  heat  and  cold. 

It  is  found  by  experience,  that  all  bodies  expand  by  heat, 
and  contract  by  cold  ; and  hence  the  degrees  of  expansion 
become  the  measure  of  the  degrees  of  heat.  Fluids  are 
more  convenient  for  this  purpose  than  solids  ; and  quick" 
silver  is  now  most  commonly  ased  for  it  A very  fine  glass 
ube,  having  a preUy  iarge  hollow  ball  at  the  bottom,  is  . 
illed  about  half  way  up  with  quicksilver  : the  whole  being 
.hen  heated  very  hot  till  the  quicksilver  rise  quite  to 
he  top,  the  top  is  then  hermetically  sealed,  so  as  perfectly 
o exclude  all  communication  with  the  outward  air  Then, 
n cooling,  the  quicksilver  contracts,  and  consequently  its 
urface  descends  in  the  tube,  till  it  come  to  a certain  point, 
orrespondent  to  the  temperature  or  heat  of  the  air.  And 
vhen  the  weather  becomes  warmer,  the  quicksilver  expands, 
Vol.  II.  31  and 


234 


THE  THERMOMETER. 


and  its  surface  rises  in  the  tube  ; and 
again  contracts  and  descends  when  the 
weather  becomes  cooler.  So  that,  by 
placing  a scale  of  any  divisions  against 
the  side  of  the  tube,  it  will  show  the 
degrees  of  heat  by  the  expansion  and 
contraction  of  the  quicksilver  in  the 
tube  ; observing'  at  what  division  of  the 
Scale  the  top  of  the  quicksilver  stands. 

And  the  method  of  preparing  the  scale, 
as  used  in  England,  is  thus  : — Bring  the 
thermometer  into  the  temperature  of 
freezing,  by  immersing  the  ball  in  water 
just  freezing,  or  in  ice  just  thawing,  and 
mark  the  scale  where  the  mercury  then 
stands,  for  the  point  of  freezing.  Next, 
immerge  it  in  boiling  water  ; and  the 
quicksilver  will  rise  to  a certain  height 
in  the  tube  ; which  mark  also  on  the 
scale  for  the  boiling  point,  or  the  heat 
of  boiling  water.  Then  the  distance  be- 
tween these  two  points,  is  divided  into 
180  equal  division,  or  degrees  ; and  the 
like  equal  degrees  are  also  continued  to  any  extent  below  the 
freezing  point,  and  above  the  boiling  point.  The  divisions 
are  then  numbered  as  follows  ; namely  at  the  freezing  point 
is  set  the  number  32,  and  consequently  212  at  the  boiling 
point ; and  all  the  other  numbers  in  their  order. 

This  division  of  the  scale  is  commonly  called  Fahrenheit's 
According  to  this  division,  55  is  at  the  mean  temperature  of 
the  air  in  this  country  ; and  it  is  in  this  temperature,  and  in 
an  atmosphere  which  sustains  a column  of  30  inches  of 
quicksilver  in  the  barometer  that  all  measures  and  specific 
gravities  are  taken,  unless  when  otherwise  mentioned;  and 
in  this  temperature  and  pressure  the  relative  weights,  or 
specific  gravities  of  air,  water,  and  quicksilver,  are  as 

If  for  air,  C and  these  also  are  the  weights  of  a cu- 

1000  for  water,  < bic  foot  of  each,  in  avoirdupois  ounces 
13600  for  mercury  ; f in  that  state  of  the  barometer  anc 
thermometer.  For  other  states  of  the  thermometer,  eacl 
of  these  bodies  expands  or  contracts  according  to  the  follow 
ing  rate,  with  each  degree  of  heat,  viz. 

Air  about  - part  of  its  bulk, 

Water  about  e sW Part  of  its  bulk, 

Mercury  about  9eVoPart  of  its  bulk 


0.' 


I 235  J 

ON  THE  MEASUREMENT  OF  ALTITUDES  BY  THE 

barometer  and  thermometer. 

391.  FROM  the  principles  laid  down  in  the  scholium  to 
prop  76.  concerning  the  measuring  of  altitudes  by  the  baro- 
im  ter,  and  the  foregoing  descriptions  of  the  barometer  and 
thermometer,  we  may  now  collect  together  the  precepts  for 
the  practice  of  such  measurements,  which  are  as  follow  : 


First.  Observe  the  height  of  the  barometer  at  the  bottom 
of  any  height,  or  depth,  inteuded  to  be  measured  ; with  the 
temperature  of  the  quicksilver,  by  means  of  a thermometer 
attached  to  the  barometer,  and  also  the  temperature  of  the 
air  in  the  shade  by  a detached  thermometer. 

Secondly.  Let  the  same  thing  be  done  also  at  the  top  of  the 
said  height  or  depth,  and  at  the  same  time,  or  as  near  the 
same  time  as  may  be.  And  let  those  altitudes  of  barometer 
be  reduced  to  the  same  temperature,  if  it  be  thought  neces- 
sary, by  correcting  either  the  one  or  the  other,  that  is,  aug- 
ment the  height  of  the  mercury  in  the  colder  temperature, 
or  diminish  that  in  the  warmer,  by  its  o'  Part  f°r  every  de- 
gree of  difference  of  the  two 

Thirdly  Take  the  difference  of  the  common  logarithms 
of  the  two  heights  of  the  barometer,  corrected  as  above  if 
necessary,  cutting  off  3 figures  next  the  right  hand  for 
decimals,  when  the  log-tables  go  to  7 figures,  or  cut  off  only 
2 figures  when  the  tables  go  to  6 places,  and  so  on  ; or  in 
general  remove  the  decimal  point  4 places  more  towards  the 
right  hand,  those  on  the  left  hand  being  fathoms  in  whole 
numbers. 


Fourthly.  Correct  the  number  lajit  found  for  the  difference 
of  temperature  of  the  air,  as  follows  ; Take  half  the  sum  of 
the  two  temperatures,  for  the  mean  one  : and  for  every  de- 
gree which  this  differs  from  the  temperature  31°,  take  so 
many  times  the  part  of  the  fathoms  above  found,  and 
add  them  if  the  mean  temperature  be  above  31°,  but  subtract 
them  if  the  mean  temperature  be  below  31°  ; and  the  sum  or 
difference  will  be  the  true  altitude  in  fathoms  : or,  being 
multiplied  by  6,  it  will  be  the  altitude  in  feet. 


392.  Example  1.  Let  the  state  of  the  barometers  and 
thermometers  be  as  follows  ; to  find  the  altitude,  viz. 


Ans.  the  alt.  is 

71 9^  fathoms. 


Barom. 

Thermom. 

attach. 

detach. 

Lower29-68 

57 

57 

Upper25'28 

43 

42 

393.  Exam 


236 


THE  RESISTANCE  OF  FLUIDS,  &c. 


393.  Exam.  2.  To  find  the  altitude,  when  the  state  of  the 
barometers  and  thermometers  is  as  follows,  viz. 


Barom. 

Thermom. 

attach. 

detach. 

Ans.  the  alt.  is 

Lower29-45 

38 

31 

409T^  fathoms. 

Upper26-82 

41 

35 

or  2458  feet. 

ON  THE  RESISTANCE  OF  FLUIDS.  WITH  THEIR 
FORCES  AND  ACTIONS  ON  BODIES. 

PROPOSITION  LXXVII, 

394.  If  any  Body  Move  through  a Fluid  at  Rest , or  the  Fluid 
Move  against  the  Body  at  Rest  ; the  Force  or  Resistance  of 
the  Fluid  against  the  Body,  will  be  as  the  Square  of  the  Velo- 
city and  the  Density  of  he  Fluid.  That  is,  r a dv2 . 

For,  the  force  or  resistance  is  as  the  quantity  of  matter 
or  particles  struck,  and  the  velocity  with  which  they  are 
struck.  But  the  quantity  or  number  of  particles  struck  in 
any  time,  are  as  the  velocity  and  the  density  of  the  fluid. 
Therefore  the  resistance,  or  force  of  the  fluid,  is  as  the  den- 
sity and  square  of  the  velocity. 

395.  Corol.  1.  The  resistance  to  any  plane,  is  also  more 
or  less,  as  the  plane  is  greater  or  less  ; and  therefore  the 
resistance  on  any  plane,  is  as  the  area  of  the  plane  a,  the 
density  of  the  medium,  and  the  square  of  the  velocity.  That 
is,  r « adv2 . 

396.  Corol.  2.  If  the  motion  be  not  perpendicular,  but 
oblique  to  the  plane,  or  to  the  face  of  the  body  ; then  the 
resistance,  in  the  direction  of  motion,  will  be  diminished  in 
the  triplicate  ratio  of  radius  to  the  sine  of  the  angle  of  in- 
clination of  the  plane  to  the  direction  of  the  motion,  or  as 
the  cube  of  radius  to  the  cube  of  the  sine  of  that  angle.  So 
that  r cc  adv2s 3,  putting  1 = radius,  and  s = sine  of  the 
angle  of  inclination  cab. 

For,  if  ab  be  the  plane,  ac  the 
direction  of  motion,  and  bc  perpen- 
dicular to  ac  ; then  no  more  particles 
meet  the  plane  than  what  meet  the 
perpendicular  bc,  and  therefore  their 
number  is  diminished  as  ab  to  bc  or 
as  1 to  s.  But  the  force  of  each  par- 


ticle. 


THE  RESISTANCE  OF  FLUIDS,  &c. 


237 


tide,  striking  the  plane  obliquely  in  the  direction  ca,  is 
also  diminished  as  ab  to  ec,  or  as  1 to  s ; therefore  the 
resistance,  which  is  perpendicular  to  the  face  of  the  plane 
by  art  52,  is  as  l2  to  s2.  But  again,  this  resistance  in  the 
direction  perpendicular  to  the  face  of  the  plane,  is  to  that 
in  the  direction  ac,  by  art.  51,  as  ab  to  bc,  or  as  1 to  s. 
Consequently,  on  all  these  accounts,  the  resistance  to  the 
plane  when  moving  perpendicular  to  its  face,  is  to  that 
when  moving  obliquely,  as  l3  to  s®,  or  1 to  s3.  That  is, 
the  resistance  in  the  direction  of  the  motion,  is  diminished 
as  1 to  s3,  or  in  the  triplicate  ratio  of  radius  to  the  sine  of 
inclination. 

PROPOSITION  LXXVIII. 


397.  The  Real  Resistance  to  a Plane,  by  a Fluid  acting  in  a 
Direction  perpendicular  to  its  Face,  is  equal  to  the  Weight 
of  a Column  of  the  Fluid,  whose  Base  is  the  Plane,  and  Al- 
titude equal  to  that  which  is  due  to  the  Velocity  of  the  Mo- 
tion, or  through  which  a Heavy  Body  must  fall  to  acquire 
that  Velocity. 


The  resistance  to  the  plane  moving  through  a fluid,  is 
the  same  as  the  force  of  the  fluid  in  motion  with  the  same 
velocity,  on  the  plane  at  rest.  But  the  force  of  the  fluid  in 
motion,  is  equal  to  the  weight  or  pressure  which  generates 
that  motion  ; and  this  is  equal  to  the  weight  or  pressure  of  a 
column  of  the  fluid,  whose  base  is  the  area  of  the  plane,  and 
its  altitude  that  which  is  due  to  the  velocity. 

398.  Corol.  1.  If  a denote  the  area  of  the*  plane,  v the 
velocity,  n the  density  or  specific  gravity  of  the  fluid,  and 
g — 16yL  feet,  or  193  inches.  Then  the  altitude  due  to 


the  velocity  v being  — , therefore  a X n X — = ——will 
J ° 4 g 4g  4, g 

be  the  whole  resistance,  or  motive  force  r. 

399.  Corol.  2.  If  the  direction  of  motion  be  not  perpen- 
dicular to  the  face  of  the  plane,  but  oblique  to  it,  in  any 
angle,  whose  sine  is  s.  Then  the  resistance  to  the  plane  will 


be 


anv2  ss 
4 g 

400.  Corol. 


3.  Also,  if  w denote  the  weight  of  the  body, 
whose  plane  face  a is  resisted  by  the  absolute  force  r ; then 

the  retarding  force  f,  or  - will  be  . 

J w 4 gw 

401.  Cqrol.  4.  And  if  the  body  be  a cylinder,  whose  face 

or 


238 


THE  RESISTANCE  OF  FLUIDS,  &c. 


or  end  is  a,  and  radius  r,  moving  in  the  direction  of  its  axis  , 
because  then  s = 1,  and  a = pr2 , where  p = 3-1416  ; then 

^ r-  will  be  the  resisting  force  r,  and  the  retarding 

force  /. 

402  Corol  5.  This  is  the  value  of  the  resistance  when 
the  end  of  the  cylinder  is  a plane  perpendicular  to  its  axis, 
or  to  the  direction,  of  motion.  But  weie  its  face  an  elliptic 
section,  or  a conical  surface,  or  any  other  figure  every  where 
equally  inclined  to  the  axis,  or  dire  tioo  of  motion,  the  sine 
or  inclination  beings:  then,  the  number  of  particles  of  the 
fluid  striking  the  face  being  still  the  same,  but  the  force  of 
each  opposed  to  the  direction  of  motion,  diminished  in  the 
duplicate  ratio  of  radius  to  the  sine  of  inclination,  the  resist^ 

ing  force  r would  be 

PROPOSITION  LXXIX. 

403.  The  Resistance  to  a Sphere  moving  through  a Fluid , is 
but  Half  the  Resistance  to  its  Great  Circle,  or  to  the  End 
of  a Cylinder  of  the  same  Diameter,  moving  with  an  Equal 
Velocity. 

Let  afeb  be  half  the  sphere,  moving 
in  the  direction  ceg.  Describe  the  para- 
boloid aiekb  on  the  same  base.  Let  any 
particle  of  the  medium  meet  the  semicir- 
cle in  f,  to  which  draw  the  tangent  fg, 
the  radius  Fc,-and  the  ordinate  fih.  Then 
the  force  of  any  particle  on  the  surface  at 
y,  is  to  its  force  on  the  base  at  h,  as  the 
square  of  the  sine  of  the  angle  g,  or  its 
equal  the  angle  fch,  to  the  square  of  radius,  that  is,  as 
hf3  to  cf2.  Therefore  the  force  of  all  the  particles,  or  the 
whole  fluid,  on  the  whole  surface,  is  to  its  force  on  the 
circle  of  the  base,  as  all  the  hf2  to  as  many  times  cf2. 
But  cf2  is  = ca2  = ac  . cb,  and  hf2  = ah  . hb  by  the 
nature  of  the  circle  : also,  ah  . hb  : ac  . cb  : : hi  : cf.  by 
the  nature  of  the  parabola  : consequently  the  force  on  the 
spherical  surface,  is  to  the  force  on  its  circular  base,  as 
all  the  hi’s  to  as  many  ce’s,  that  is,  as  the  content  of  the 
paraboloid  to  the  content  of  its  circumscribed  cylinder,  namely, 
as  1 to  2. 

404.  Corol.  Hence,  the  resistance  to  the  sphere  is  r = 
being  the  half  of  that  of  a cylinder  of  the  same 

8ff  ® 

diameter. 


SPECIFIC  GRAVITY. 


239 


diameter.  For  example,  a 91b  iron  ball,  whose  diameter  is 
4 inches,  when  moving  through  the  air  with  a velocity  of 
1600  feet  per  second,  would  meet  a resistance  which  is  equal 
to  a weight  of  132|lb,  over  and  above  the  pressure  of  the  at- 
mosphere, for  want  of  the  counterpoise  behind  the  wall. 

PRACTICAL  EXERCISES  CONCERNING  SPECIFIC 
GRAVITY. 

The  Specific  Gravities  of  Bodies  are  their  relative  weights 
contained  under  the  same  given  magnitude  ; as  a cubic  foot, 
or  a cubic  inch,  &c. 

The  specific  gravities  of  several  sorts  of  matter,  are  ex- 
pressed by  the  numbers  annexed  to  their  names  in  the  Table 
of  Specific  Gravities,  at  page  211  ; from  which  the  numbers 
are  to  be  taken,  when  wanted. 

Note.  The  several  sorts  of  wood  are  supposed  to  be  dry. 
Also,  as  a cubic  foot  of  water  weighs  just  1000  ounces  avoit- 
dupois,  the  numbers  in  the  table  express,  net  only  the  specific 
gravities  of  the  several  bodies,  but  also  the  weight  of  a cubic 
foot  of  each  in  avoirdupois  ounces  ; and  hence,  by  proportion, 
the  weight  of  any  other  quantity,  or  the  quantity  of  any  other 
weight,  may  be  known,  as  in  the  following  problems. 

PROBLEM  I. 

To  find  the  Magnitude  of  any  Body,  from  its  Weight. 

As  the  tabular  specific  gravity  of  the  body, 

Is  to  its  weight  in  avoirdupois  ounces, 

So  is  one  cubic  foot,  or  1728  cubic  inches, 

To  its  content  in  feet,  or  inches,  respectively. 

EXAMPLES. 

Exam  1.  Required  the  content  of  an  irregular  block  of 
common  stone,  which  weights  1 cwt  or  1121b. 

Ans.  1228|  cubic  inches. 

Exam,  2.  How  many  cubic  inches  of  gunpowder  are  there 
in  lib  weight  ? Ans.  29|  cubic  inches  nearly. 

Exam.  3.  How  many  cubic  feet  are  there  in  a ton  weight  of 
dry  oak  ? Ans.  38iff  cubic  feet. 

PROBLEM 


240 


SPECIFIC  GRAVITY. 


PROBLEM  H. 

To  find  the  Weight  of  a Body  from  its  Magnitude . 

As  one  cubic  foot,  or  1728  cubic  inches, 

Is  to  the  content  of  the  body, 

So  is  its  tabular  specific  gravity, 

To  the  weight  of  the  body. 

EXAMPLES. 

Exam.  1.  Required  the  weight  of  a block  of  marble,  whose 
length  is  63  feet,  and  breadth  and  thickness  each  12  feet  ; 
being  the  dimensions  of  one  of  the  stones  in  the  walls  of  Bal- 
beck  ? 

Ans.  683x\  ton,  which  is  nearly  equal  to  the  burden 
of  an  East-India  ship. 

Exam.  2.  What  is  the  weight  of  1 pint,  ale  measure,  of 
gunpowder  ? Ans.  19  oz  nearly. 

Exam.  3.  What  is  the  weight  of  a block  of  dry  oak.  which 
measures  10  feet  in  length,  3 feet  broad,  and  24  feet  deep  ; 

'Ans.  4335i|Ib. 

PROBLEM  IIL 

To  find  the  Specific  Gravity  of  a Body. 

Case  1.  When  the  body  is  heavier  than  water,  weigh  it 
both  in  water  and  out  of  water,  and  take  the  difference,  which 
will  be  the  weight  lost  in  water.  Then  say, 

As  the  weight  lost  in  water, 

Is  to  the  whole  weight. 

So  is  the  specific  gravity  of  water. 

To  the  specific  gravity  of  the  body. 

EXAMPLE. 

A piece  of  stone  weighed  10lb,  but  in  water  only  6£lb,  re- 
quired its  specific  gravity  ? Ans.  2609. 

Case  2.  When  the  body  is  lighter  than  water;  so  that  it  will 
not  quite  sink,  affix  to  it  a piece  of  another  body,  heavier  than 
water,  so  that  the  mass  compounded  of  the  two  may  sink  to- 
gether. Weigh  the  denser  body  and  the  compound  mass  se- 
parately, both  in  water  and  out  of  it ; then  find  how  much  each 
loses  in  water,  by  subtracting  its  weight  in  water  from  its 
weight  in  air  ; and  subtract  the  less  of  these  remainders  from 
the  greater.  Then  say, 

As 


SPECIFIC  GRAVITY. 


241 


As  the  last  remainder. 

Is  to  the  weight  of  the  light  body  in  air, 

Sp  is  the  specific  gravity  of  water, 

To  the  specific  gravity  of  the  body. 

EXAMPLE. 

Sppose  a piece  of  elm  weighs  151b  in  air  ; and  that  a piece 
■pper  which  weighs  18lb  in  air,  and  161b  in  water,'  is  affix~ 
it,  and  that  the  compound  weighs  61b  in  water  ; required 
; lecific  gravity  of  the  elm  ? Ans.  600. 

PROBLEM  IV. 

i jid  the  Quantities  of  Two  Ingredients,  in  a Given  Compound . 

Ike  the  . three  differences  of  every  pair  of  the  three 
, ;<5c  gravities,  namely,  the  specific  gravities  of  the  com- 
ul  and  each  ingredient  ; and  multiply  the  difference  of 
, i,  two  specific  gravities  by  the  third.  Then  say,  as  the 
»?st  product,  is  to  the  whole  weight  of  the  compound, 

' leach  of  the  other  products,  to  the  two  weights  of  the  in- 
!(3DtS. 

EXAMPLE. 

Composition  of  1121b  being  made  of  tin  and  copper, 
in);  specific  gravity  if  found  to  be  8784  ; required  the 
kfty  of  each  ingredient,  the  specific  gravity  of  tin  being 
and  of  copper  9000  ? 

Ans.  there  is  1001b  of  copper  > . .,  ... 

! and  consequently  121b  of  tin  \ mthe  C0mP0Sltl0n‘ 


■ HE  WEIGHT  AND  DIMENSIONS  OF  BALLS  AND 
SHELLS. 


T : weight  and  dimensions  of  Balls  and  Shells  might  be 
ur  from  the  problems  last  given,  concerning  specific  gra- 
J.  But  they  may  be  found  still  easier  by  means  of  the 
pemented  weight  of  a ball  of  a given  size,  from  the 
'Ov  proportion  of  similar  figures,  namely,  as  the  cubes 
th  r diameters. 

PROBLEM  I. 

V find  the  Weight  of  an  Iron  Ball,  from  its  Diameter. 

Aiiron  ball  of  4 inches  diameter  weighs  9lb,  and  the 
eigs  being  as  the  cubes  of  the  diameters,,  it  will  be,  as  64 
V<j.  II.  32  (which 


BALLS  AND  SHELLS. 


24  2 

(which  is  the  cube  of  .4)  is  to  9 its  weight,  so  is  the  cut  o( 
the  diameter,  of  any  other  ball,  to  its  weight.  Or.  take 
the  cube  of  the  diameter,  for  the  weight.  Or,  take  } olia 
cube  of  the  diameter,  and  a of  that  again,  and  add  the  ?» 
together,  for  the  weight. 

EXAMPLES. 

Exam.  1.  The  diameter  of  an  iron  shot  being  6-7  in<s,l 
required  its  weight  ? Ans.  42-2  b.1 

Exam.  2.  What  is  the  weight  of  an  iron  ball,  whose** 
meter  is  5’54  inches  ? Ans.  24lb  ne  j. 

PROBLEM  II. 

To  find  the  Weight  of  a Leaden  Ball. 

A leaden  ball  of  one  inch  diameter  weighs  T\,  of  a lb  ; tin 
fore  as  the  cube  of  1 is  to  T\  or  as  14  is  to  3,  so  is  the  M 
of  the  diameter  of  a leaden  ball,  to  its  weight.  Or,  take  S 
the  cube  of  the  diameter,  for  the  weight,  nearly. 

EXAMPLES. 

E^am.  1.  Required  the  weight  of  a' leaden  ball  of  6-6  ina 
diameter?  Ans.  6 1 *6 Ifc 

Exam.  2.  What  is  the  weight  of  a leaden  ball  of  5-30 iim 
diameter?  Ans.  321b nelf 

PROBLEM  in. 

To  find  the  Diameter  of  an  Iron  Ball. 

Multiply  the  weight  by  and  the  cube  root  of  ther* 
duct  will  be  the  diameter. 

EXAMPLES. 

Exam.  1.  Required  the  diameter  of  a 42lb  iron  ball?  a 

Ans.  6'685inee 

Exam.  2.  What  is  the  diameter  of  a 24lb  iron  ball  ? 

Ans.  5'54  inie- 

PROBLEM  TV. 

To  find  the  Diameter  of  a Leaden  Ball. 

Multiply  the  weight  by  14,  and  divide  the  produi 
3 ; then  the  cube  root  of  the  quotient  will  be  the  diamet'- 

EXAM-ES 


BALLS  AND  SHELLS. 


243 


EXAMPLES. 


xam.  1.  Required  the  diameter  of  a 641b  leaden  ball  ? 

Ans.  6’684  inches. 

aam.  2.  What  is  the  diameter  of  an  81b  leaden  ball  ? 

Ans.  3‘343  inches. 

PROBLEM  V. 

To  find  the  Weight  of  an  Iron  Shell. 
k|  I4ke  of  the  difference  of  the  cubes  of  the  external  and 
itjtnal  diameter,  for  the  weight  of  the  shell, 
mat  is,  from  the  cube  of  the  external  diameter,  take  the 
nl  of  the  internal  diameter,  multiply  the  remainder  by  9, 
lunivide  the  product  by  64. 


EXAMPLES. 

The  outside  diameter  of  an  iron  shell  being  12-8, 
; diameter  9 1 inches  ; required  its  weight  ? 

Ans.  188  9411b. 
1:am.  2.  What  is  the  weight  of  an  iron  shell,  whose  ex- 
1 and  internal  diameters  are  9-8  and  7 inches  ? 

Ans.  84ilb. 

PROBLEM  VI. 

To  find  how  much  Powder  will  fill  a Shell. 
vide  the  cube  of  the  internal  diameter,  in  inches,  by 
for  the  lbs  of  powder* 

EXAMPLES. 

I am.  1.  How  much  powder  will  fill  the  shell  whose  in- 
;rnl  diameter  is  9‘1  inches  ? Ans.  13T2^lb  nearly. 

I am.  2.  How  much  powder  will  fill  a shell  whose  in- 


sn  l diameter  is  7 inches  ? 


Ans.  61b. 


PROBLEM  VII. 


I 


or 


To  find  how  much  Powder  will  fill  a Rectangular  Box. 
jd  the  content  of  the  box  ill  inches,  by  multiplying  the 
l,  breadth,  and  depth  all  together.  Then  divide  by  30 
e pounds  of  powder, 


EXAMPLES. 

Eam.  1.  Required  the  quantity  of  powder  that  will  fill 
i he,  the  length  being  15  inches,  the  breadth  12,  and  the 
lep  10  inches  ? Ans.  60lb. 

EXAM,  e 


244 


POWDER  AND  SHELLS,  &c 


Exam.  2.  How  much  powder  will  fill  a cubical  box  wh  e 
side  is  12  inches  ? Ans.  57). 

PROBLEM  VIII. 

To  find  how  much  Powder  will  Jill  a Cylinder. 
Multiply  the  square  of  the  diameter  by  the  length,  to; 
divide  by  38'2  for  the  pounds  ol  powder. 

EXAMPLES. 

Exam.  1.  How  much  powder  will  the  cylinder  hold,  wIp  I 
diameter  is  10  inches,  and  length  20  inches  '!  Ans  5’  !b  neaM 
Exam.  2.  How  much  powder  can  be  contaiued  in  e I 
cylinder  whose  diameter  is  4 inches,  and  iength  12  inches  4 

Ans.  5y|l 

PROBLEM  IX. 

To  Jind  the  Size  of  a Shell  to  contain  a Given  Weight  of  Po m| 
Multiply  the  pounds  of  powder  by  ST'S,  and  the  ce 
root  of  the  product  will  be  the  diameter  in  inches. 

EXAMPLES. 

• • ■ - » J 

Exam.  1.  What  is  the  diameter  of  a shell  that  will  Up 
13£  of  powder  ? Ans.  9-1  inc-. 

Exam.  2.  What  is  the  diameter  of  a shell  to  contain b , 
of  powder  ? Ans.  7 inc  '<■  I 

PROBLEM  X. 

To  find  the  Size  of  a Cubical  Box  to  contain  a given  Weigh  i 
Powder. 

Multiply  the  weight  in  pounds  by  SO,  and  the  cubei! 
of  the  product  will  be  the  side  of  the  box  in  inches. 

EXAMPLES.  , 

Exam  1.  Required  the  side  of  a cubical  box,  to  hold  cb 
of  gunpowder  ? • Ans.  11-44  inch- 

Exam.  2.  Required  the  side  of  a cubical  box.  to  Id 

4001b  of  gunpowder  ? Ans.  22-89  inch- 

PROBLEM  XL 

To  find  what  Length  of  a Cylinder  will  be  filled  by  a gi* 
Weight  of  Gunpowder. 

Multiply  the  weight  in  pounds  by  38-2,  and  divide  e 

product  by  the  square  of  the  diameter  in  inches,  for  e 

length.  EXAMPlp- 


PILING  OF  BALLS  AND  SHELLS. 


2 4 r> 


EXAMPLES. 

Exam.  1.  What  length  of  a 36-pounder  gun,  of  6f  inches 
diameter,  will  be  filled  with  121b  of  gunpowder. 

Ans.  10-314  inches 

Exam.  2.  WJiat  length  of  a cylinder,  of  8 inches  diameter, 
may  be  filled  with  -201b  of  powder  ? Ans.  1 !}£  inches 


OF  THE  PILING  OF  BALLS  AND  SHELLS. 

Iron  Balls  and  Shells  are  commonly  piled  by  horizontal 
courses,  either  in  a pyramidical  or  in  a wedge-like  form  ; the 
base  being  either  an  equilateral  triangle,  or  a square,  or  a 
rbctangle.  In  the  triangle  and  square,  the  pile  finishes  in  a 
single  ball  ; but  in  the  rectangle  it,  finishes  in  a single  row  of 
balls,  like  an  edge. 

In  triangular  and  square  piles,  the  number  of  horizontal 
rows,  or  courses,  is  always  equal  to  the  number  of  balls  in 
one  side  of  the  bottom  row.  And  in  rectangular  piles,  the 
number  of  rows  is  equal  to  the  number  of  balls  in  the  breadth 
of  the  bottom  row.  Also,  the  number  in  the  top  row,  or  edge, 
is  one  more  than  the  difference  between  the  length  and  breadth 
of  the  bottom  row. 

9 

PROBLEM  I. 


To  find  the  number  of  Balls  in  a Triangular  Pile. 


Multiply  continually  together  the  number  of  balls  in  one 
sidelof  the  bottom  row,  and  that  number  increased  by  f , also  the 
same  number  increased  by  2 ; then  } of  the  last  product  will 
be  the  answer. 


That  is, 


n -J-  1 • n + 2 . 


is  the  number  or  sum,  where  n is 


the  number  in  the  bottom  row. 


EXAMPLES. 


Exam.  1.  Required  the  number  of  balls  in  a triangular  pile  , 
each  side  of  the  base  containing  30  balls  ? Ans.  4960. 

Exam.  2.  How  many  balls  are  in  the  triangular  pile,  each 
side  of  the  base  containing  20  ? Ans.  1540. 


PROBLEM 


246 


PILING  OF  BALLS  AND  SHELLS. 


PROBLEM  11. 

To  find  the  Number  of  Balls  in  a Square  Pile. 

Multiply  continually  together  the  number  in  one  side 
of  the  bottom  course,  that  number  increased  by  1,  and  double 
the  same  number  increased  by  1 ; then  £ of  ihe  last  product 
will  be  the  answer. 

That  is,  U ' n ^ ‘2r—~  is  the  number. 

EXAMPLES. 

Exam.  1.  How  many  balls  are  in  a square  pile  of  30  rows  ? 

Ans.  9455. 

Exam.  2.  How  many  balls  are  in  a square  pile  of  20  rows  ? 

Ans.  2870. 

PROBLEM  1IL 

To  find  the  Number  of  Balls  in  a Rectangular  Pile. 

From  3 times  the  number  in  the  length  of  the  base  row 
subtract  one  less  than  the  breadth  of  the  same,  multiply  the 
remainder  by  the  same  breadth,  and  the  product  by  one  more 
than  the  same,  and  divide  by  6 for  the  answer. 

That  is,  b ' b Alhl  is  the  number  ; where  l is 

6 

the  length,  and  b the  breadth  of  the  lowest  course. 

Note.  In  all  the  piles  the  breadth  of  the  bottom  is  equal  to 
the  number  of  courses.  And  in  the  oblong  or  rectangular 
pile,  the  top  row  is  one  more  than  the  difference  between  the 
length  and  breadth  of  the  bottom. 

BXAMPLES.  • 

Exam.  1.  Required  the  number  of  balls  in  a reclangulai 
pile,  the  length  and  breadth  of  the  base  row  being  46  and  15  ? 

Ans.  4960. 

Exam.  2.  How  manjr  shot  are  in  a rectangular  complete 
pile,  the  length  of  the  bottom  course  being  59,  and  its  breadth 
20?  Ans.  11060. 


PROBLEM  IV. 

To  find  the  Number  of  Balls  in  an  Incomplete  Pile. 

From  the  number  in  the  whole  pile,  considered  as  com- 
plete, subtract  the  number  in  the  upper  pile  which  is  want- 


DISTANCES  BY  SOUND. 


247 


mg  at  the  top,  both  computed  by  the  rule  for  their  proper  form ; 
and  the  remainder  will  be  the  number  in  the  frustum,  or  in- 
complete pile. 

EXAMPLES. 

Exam.  1.  To  find  the  number  of  shot  in.  the  incomplete 
triangular  pile,  one  side  of  the  bottom  course  being  40,  and 
the  top  course  20  ? Ans  10150. 

Exam.  2.  How  many  shot  are  in  the  incomplete  triangular 
pile,  the  side  of  the  base  being  24,  and  of  the  top  8 ? 

Ans.  2516. 

Exam.  3.  How  many  balls  are  in  the  incomplete  square 
pile,  the  side  of  the  base  being  24,  and  of  the  top  8 ? 

Ans.  4760. 

Exam  4.  How  many  shot  are  in  the  incomplete  rectangular 
pile,  of  12  courses,  the  length  and  breadth  of  the  base  being 
40  and  20  ? Ans.  614G. 


OF  DISTANCES  BY  THE  VELOCITY  OF  SOUND. 

By  various  experiments  it  has  been  found,  that  sound  flies  ; 
through  the  air,  uniformly  at  the  rate  of  about  1142  feet  in 
1 second  of  time,  or  a mile  in  4f  or  seconds.  And  there- 
fore, by  proportion,  any  distance  may  be  found  corresponding 
to  any  given  time  ; namely,  multiplying  the  given  time,  in 
secoqds,  by  1142,  for  the  corresponding  distance  in  feet  ; 
or  taking  T\  of  the  given  time  for  the  distance  in  miles.  Or 
dividing  any  given  distance  by  these  numbers,  to  find  the  cor- 
responding time. 

Note.  The  time  for  the  passage  of  sound  in  the  interval  be- 
tween seeing  the  flash  of  a gun,  or  lightning,  and  hearing 
the  report,  may  be  observed  by  a watch,  or  a small  pendulum. 
Or,  it  may  be  observed  by  the  beats  of  the  pulse  in  the  wrist, 
countiug,  on  an  average,  about  70  to  a minute  for  persons  in 
moderate  health,  or  5i  pulsations  to  a mile  : and  more  or  less 
according  to  circumstances. 

EXAMPLES. 

Exam.  1.  After  observing  a flash  of  lightning,  it  was  12 
seconds  before  the  thunder  was  heard  ; required  the  distance 
of  the  cloud  from  whence  it  came  ? Ans.  24  miles. 

Exam.  2.  How  long,  after  firing  the  Tower  guns,  may 

the 


243 


PRACTICAL  EXERCISES 


the  report  be  heard  at  Shooter’s-Hill,  supposing  the  distance 
to  be  8 miles  in  a straight  line  ? Ans.  374  seconds 

Exam.  3.  After  observing  the  firing  of  a large  cannon  at  a 
distance,  it  was  7 seconds  before  the  report  was  heard  ; what 
was  its  distance  ? Ans.  ]i  mile. 

Exam.  4.  Perceiving  a man  at  a distance  hewing  down  a 
tree  with  an  axe,  I remarked  that  6 of  my  pulsations  passed 
between  seeing  him  strike  and  hearing  the  report  of  the  blow  ; 
what  was  the  distance  between  us,  allowing  70  pulses  to  a 
minute  ? Ans.  1 mile  and  J 98  yards 

Exam.  5.  How  far  off  was  the  cloud  from  which  thunder 
issued,  whose  report  was  5 pulsations  after  the  flash  of  light- 
ning ; counting  75  to  a minute  ? Ans.  1523  yards. 

Exam.  6.  If  I see  the  flash  of  a cannon,  fired  by  a ship  in 
distress  at  sea,  and  hear  the  report  33  seconds  after,  how  far 
is  she  off?  Ans.  7T'T  miles 


PRACTICAL  EXERCISES  IN  MECHANICS,  STATICS 

HYDROSTATICS,  SOUND,  MOTION,  GRAVITY,  PRO- 
JECTILES, AND  OTHER  BRANCHES  OF  NATURAL 
PHILOSOPHY. 

Question  1.  Required  the  weight  of  a cast  iron  ball  of 
3 inches  diameter,  supposing  the  weight  of  a cubic  inch  of  the 
metal  to  be  0-2581b  avoirdupois  ? Ans.  3-647391b. 

Quest.  2.  To  determine  the  weight  of  a hollow  spherical 
iron  shell,  5 inches  in  diameter,  the  thickness  of  the  metal  be- 
ing one  inch  ? Ans.  13-2387lb. 

Quest.  3.  Being  one  day  ordered  to  observe  how  far  a bat 
tery  of  cannon  was  from  me,  I counted,  by  my  watch,  17  sec 
onds  between  the  time  of  seeing  the  flash  and  hearing  the  re- 
port ; what  then  was  the  distance  ? Ans  3f  miles. 

Quest.  4.  It  is  proposed  to  determine  the  proportional 
quantities  of  matter  in  the  earth  and  moon  ; the  density  of  the 
former  being  to  that  of  the  latter,  as  10  to  7,  and  their  diame- 
ters as  7930  to  2160.  Ans.  as  71  to  1 nearly. 

Quest.  5.  What  difference  is  there,  in  point  of  weight, 
between  a block  of  marble,  containing  1 cubic  foot  and  a half 
and  another  of  brass  of  the  same  dimensions  ? 

Ans.  4961b  14oz. 

Quest.  6.  In  the  walls  of  Balbeck  in  Turkey,  the  ancient 
Heliopolis,  there  are  three  stones  laid  end  to  end,  now  in  sight. 

* that 


IN  NATURAL  PHILOSOPHY. 


249 


that  measure  in  length  61  yards  ; one  of  which  in  particular 
is  21  yards  or  63  feet  long,  12  feet  thick,  and  12  feet  broad  : 
now  if  this  block  be  marble,  what  power  would  balance  it,  so 
as  to  prepare  it  for  moving  ? 

Ans.  683T7g  tons,  the  burden  of  an  East-India  ship. 

Quest.  7.  The  battering-ram  of  Vespasian  weighed,  sup- 
pose 10,000  pounds ; and  was  moved,  let  us  admit,  with 
such  a velocity,  by  strength  of  hand,  as  to  pass  through 
20  feet  in  one  second  of  time  ; and  this  was  found  sufficient 
to  demolish  the  walls  of  Jerusalem.  The  question  is,  with 
what  velocity  a 32lb  ball  must  move,  to  do  the  same  execu- 
tion ? Ans.  6250  feet. 

Quest.  8.  There  are  two  bodies,  of  which  the  one  con- 
tains 25  times  the  matter  of  the  other,  or  is  25  times  heavier  ; 
hut  the  less  moves  with  1000  times  the  velocity  of  the  great- 
• er  ; in  what  proportion  then  are  the  momenta,  or  forces,  with 
which  they  moved  ? 

Ans.  the  less  moves  with  a force  40  times  greater. 

Quest.  9.  A body,  weighing  20lb,  is  impelled  by  such  a 
force,  as  to  send  it  through  a 100  feet  in  a second  ; with  what 
velocity  then  would  a body  of  81b  weight  move,  if  it  were 
impelled  by  the  same  force  ? Ans.  250  feet  per  second. 

Quest.  10.  There  are  two  bodies,  the  one  of  which  weighs 
1001b,  the  other  60  ; but  the  less  body  is  impelled  by  a force 
8 times  greater  than  the  other  ; the  proportion  of  the  veloci- 
ties, with  which  these  bodies  move,  is  required  ? 

Ans.  the  velocity  of  the  greater  to  that  of  the  less,  as  3 to  40. 

Quest.  11.  There  are  two  bodies,  the  greater  contains  8 
times  the  quantity  of  matter  in  the  less,  and  is  moved  with 
a force  48  times  greater  ; the  ratio  of  the  velocities  of  these 
two  bodies  is  required  ? 

Ans.  the  greater  is  to  the  less,  as  6 to  1. 

Quest.  12.  There  are  two  bodies,  one  of  which  moves 
40  times  swifter  than  the  other  ; but  the  swifter  body  has 

I moved  only  one  minute,  whereas  the  other  has  been  in  mo- 
tion 2 hours  : the  ratio  of  the  spaces  described  by  these  two 
bodies  is  required  ? 

Ans.  the  swifter  is  to  the  slower,  as  1 to  3. 
Quest.  13.  Supposing  one  body  to  move  30  times  swiflar 
than  another,  as  also  the  swifter  to  move  12  minutes,  the 
other  only  1 : what  difference  will  there  be  between  the 
spaces  described  by  them,  supposing  the  last  has  moved  5 
feet?  Ans.  1795  feet. 

Quest.  14.  There  are  two  bodies,  the  one  of  which  has 
passed  over  50  miles,  the  other  only  5 ; and  the  first  hacl 
Von.  II.  33  moved 

H.. " .y. 


250 


PRACTICAL  EXERCISES 


moved  with  5 times  the  celerity  of  the  second  ; what  is  the 
ratio  of  the  times  they  have  been  in  describing  those  spaces  ? 

Ans.  as  2 to  1. 

Quest.  15  If  a Sever,  40  effective  inches  long,  will,  by 
a certain  power  thrown  successively  on  it,  in  13  hours, 
raise  a weight  104  feet  ; in  what  time  will  two  other  levers, 
each  18  elective  inches  long,  raise  an  equal  weight  73  feet  ? 

Ans.  10  hours  minutes. 

Quest.  10.  What  weight  will  a man  be  able  to  raise,  who 
presses  with  the  force  of  a hundred  and  a half,  on  the  end  of  ; 
an  equipoised  handspike,  100  inches  long,  meeting  with  a 
convenient  prop  exactly  7-’-  inches  from  the  lower  end  of  the  - 
machine  ? Ans.  20721b. 

Quest.  17.  A weight  of  Qlb,  laid  on  the  shoulder  of  a 
man,  is  no  greater  burden  to  him  than  its  absolute  weight, 
or  24  ounces  : what  difference  will  he  feel  between  the  said 
weight  applied  near  his  elbow,  at  12  inches  from  the  shoul- 
der, and  in  the  palm  of  his  hand,  28  inches  from  the  same  ; : 
and  Low'  much  more  must  his  muscles  then  draw  to  support 
it  at  right  angles,  that  is,  having  his  arm  stretched  right  out  ? ] 

Ans.  24lb  avoirdupois.  , 

Quest.  18.  What  weight  hung  on  at  70  inches  from  the 
centre  of  motion  of  a steel-yard  will  balance  a small  gun  of 
9 \ cwt,  freely  suspended  at  2 inches  distance  from  the  said 
centre  on  +he  contrary  side  ? Ans.  30|lb. 

Quest.  19  It  is  proposed  to  divide  the  beam  of  a steel- 
yard, or  to  find  the  points  of  division  where  the  weights  of 
1,  2,  3,  4,  &c.  lb,  on  the  one  side,  will  just  balance  a constant 
weight  of  95lb  at  the  distance  of  2 inches  on  the  other  side 
of  the  fulcrum  ; the  weight  of  the  beam  being  101b,  and  its 
whole  length  32  inches  ? 

Ans.  30,  15,  10,  H,  6,  5,  4f,  3f,  3£,  3,  2T\,  21,  &c. 

Que3t.  20.  Two  men  carrying  a burden  of  2001b  weight 
between  them  hung  on  a.  pole,  the  ends  of  which  rest  on  their 
shoulders  ; he  w much  of  this  load  is  borne  bv  each  man,  the 
weight  hanging  6 inches  from  the  middle,  and  the  whole 
length  of  the  pole  being  4 feet  ? Ans.  1251b  and  75lb. 

Quest.  21  If,  in  a pair  of  sca.les,  a body  weigh  90lb  in 
one  scale,  and  only  40ib  in  the  other  ; required  its  true 
weight,  and  the  preportion  of  the  lengths  of  the  two  arms  of 
the  balance  beam,  on  each  side  of  the  point  of  suspension  ? 

Ans.  the  weight  60lb,  and  the  proportion  3 to  2. 

Quest.  22.  To  rind  the  weight  ol  a beam  of  timber,  or 
other  body,  by  means  of  man's  owa  weight,  or  any  other 
weight.  For  instance,  a piece  of  tapering  timber,  24  feet 
long,  being  laid  over  a prop,  or  the  edge  of  another  beam, 
is  found  to  balance  itself  when  the  prop  is  13  feet  from  the 

less 


IN  NATURAL  PHILOSOPHY. 


251 

less  end  ; but  removing  the  prop  a foot  nearer  to  the  said 
end,  it  takes  a man’s  weight  of  2101b,  standing  on  the  less 
end,  to  hold  it  in  equilibrium.  Required  the  weight  of  the 
tree  ? Ana.  2520ib. 

Quest.  23.  If  ab  be  a cane  or  walking-stick,  40  inches 
long,  suspended  by  a string  sd  fastened  to  the  middle  point 
d : now  a body  being  hung  on  at  e,  6 inches  distance  from 
b,  is  balanced  by  a weight  of  21b,  hung  on  at  the  larger  end 
a ; but  removing  the  body  to  f,  one  inch  nearer  to  d,  the 
21b  weight  on  the  other  side  is  moved  to  g,  within  8 inches 
of  d,  before  the  cane  will  rest  in  equilibrio.  Required  the 
weight  of  the  body  ? Ans.  24lb. 

Quest.  24.  If  ab,  bc  be  two  inclined  planes,  of  the 
lengths  of  30  and  40  inches,  and  moveable  about  the  joint 
at  b : what  will  be  the  ratio  of  two  weights  p,  o_,  in  equi- 
librio on  the  planes,  in  all  positions  of  them  : and  wfcat  will 
be  the  altitude  bd  of  the  angle  b above  the  horizontal  plane 
ac,  when  this  is  50  inches  long  ? 

Ans.  feD  = 24  ; and  r to  q.  as  ab  to  bc,  or  as  3 to  4. 

Quest.  25.  A Jever,  of  6 feet  long,  is  fixed  at  right  angles 
in  a screw,  whose  threads  are  one  inch  asunder,  so  that  the 
lever  turns  just  once  round  in  raising  or  depressing  the  screw 
one  inch.  If  then  this  lever  be  urged  by  a weight  or  force 
of  501b,  with  what  force  will  the  screw  press  ? 

Ans.  2261941b. 

Quest.  26.  If  a man  can  draw  a weight  of  1501b.  up  the 
side  of  a perpendicular  wall,  of  20  feet  high  ; what  weight 
will  he  be  able  to  raise  along  a smooth  plank  of  30  feet  long, 
laid  aslope  from  the  top  of  the  wall  ? Ans.  2251b. 

Quest.  27.  If  a force  of  1501b  be  applied  on  the  head  of 
a rectangular  wedge,  its  thickness  being  2 inches,  and  the 
length  of  its  side  12  inches  ; what  weight  will  it  raise  or  ba- 
lance perpendicular  to  its  side  ? Ans.  9001b. 

Quest.  28:  If  a round  pillar  of  30  feet  diameter  be  raised 

on  a plane,  inclined  to  the  horizon  in  an  angle  of  759,  or  the 
shaft  inclining  15  degrees  out  of  the  perpendicular  : what 
length  will  it  bear  before  it  overset  ? 

Ans.  30  (2+^/3)  or  111-9615  feet. 

Quest.  29.  If  the  greatest  angle  at  which  a bank  of  na- 
tural earth  will  stand  be  45°  ; it  is  proposed  to  determine  what 
thickness  an  upright  wall  of  stone  must  be  made  throughout, 
just  to  support  a bank  of  12  feet  high  ; the  specific  gravity  of 
the  stone  being  to  that  of  earth,  as  5 tq  4. 

Ans.  *73  y'  or  4-29325  feet. 

Quest.  30.  If  the  stone  wall  be  made  like  a wedge,  or 
having  its  upright  section  a triangle,  tapering  to  a point  at 

top, 


252  PRACTICAL  EXERCISES 

top,  but  its  side  next  the  bank  of  earth  perpendicular  to  the 
horizon  ; what  is  its  thickness  at  the  bottom,  so  as  to  sup- 
port the  same  bank  ? Ans.  12^/j  or  5-36656  feet. 

Qwest  31.  But  if  the  earth  will  only  stand  at  an  angle  of 
30  degrees  to  the  horizontal  line  ; it  is  required  to  determine 
the  thickness  of  wah  in  both  the  preceding  cases  ? 

Ans.  the  breadth  of  the  rectangle  12,/Q  or  5-36656, 
but  the  base  of  the  triangular  bank  or  6-53667. 

Quest.  32  To  find  the  thickness  of  an  upright  rectan- 
gular vail.  necessary  to  support  a body  of  water  ; the  water 
being  10  feet  deep,  and  the  wall  12  feet  high  . also  the  spe- 
cific gravity  of  the  wall  to  that  of  the  water  as  11  to  7. 

Ans.  4-204374  feet. 

Quest.  33.  To  determine  the  thickness  of  the  wall  at  the 
bottom,  when  the  section  of  it  is  triangular,  and  the  altitudes 
as  before.  Ans.  5-1492866  feet. 

Quest.  34.  Supposing  the  distance  of  the  earth  from  the  j 
sun  to  le  96  millions  of  miles  ; I would  know  at  what  distance 
from  him  another  body  must  be  placed,  so  as  to  receive  light 
and  heat  quadruple  to  that  of  the  earth  ? 

Ans.  at  half  the  distance,  or  47k  millions. 

Quest.  35.  If  the  mean  distance  of  toe  sun  from  us  be  | 
106  of  his  diameters  how  much  hotter  is  it  at  the  surface 
of  the  sun,  than  under  our  equator  ? 

Ans.  11236  times  hotter 

Quest.  36.  The  distance  between  the  earth  and  tbe  sun 
being  accounted  95  millions  of  miles,  and  between  Jupiter  and 
the  sun  495  millions  ; the  degree  of  light  and  heat  received 
by  Jupiter,  compared  with  that  of  the  earth,  is  required  ? 

Ans.  or  Ecarly  Jy  of  the  earth’s  light  and  heat. 

Quest.  37  A certain  body  on  the  surface  of  the  earth 
weighs  a cwt,  or  1121b  ; the  question  is  whither  this  body 
must  be  carried,  that  it  may  weigh  only  101b  ? 

Ans.  either  at  3-3466  semi-diameters,  or  Jj  ef  a semi- 
diameter, from  the  centre. 

Quest.  38.  If  a body  weigh  1 pound,  or  16  ounces,  on  the 
surface  of  the  earth  ; what  will  its  weight  be  at  50  miles 
above  it,  takmg  the  earth’s  diameter  at  7930  miles  ? 

Ans.  15cz.  9fdr.  nearly. 

Ques^.  39.  Whereabouts,  in  the  line  between  the  earth 
and  moon,  is  their  common  centre  of  gravity  : supposing  the 
earth’s  diameter  to  be  7930  miles,  and  the  moon's  2160  ; also 

the 


IN  NATURAL  PHILOSOPHY. 


253 


the  density  of  the  former  to  that  of  the  latter,  as  99  to  68,  or 
as  10  to  7 nearly,  and  their  mean  distance  30  of  the  earth’s 
diameters  ? 

Ans.  at  parts  of  a diameter  from  the  earth’s  centre, 

or  parts  of  a diameter,  or  648  miles  below  the 

surface. 

Quest.  40.  Whereabouts,  between  the  earth  and  moon, 
are  their  attractions  equal  to  each  other  ? Or  where  must 
another  body  be  placed,  so  as  to  remain  suspended  in  equi- 
librio,  not  being  more  attracted  to  the  one  than  to  the  other  or 
having  no  tendency  to  fall  either  way  ? their  dimensions  being 
as  in  the,  last  question. 

Ans.  From  the  earth’s  centre  26T9T  > of  the  earth’s 
From  the  moon’s  centre  3^r  $ diameters. 

Quest.  41.  Suppose  a stone  dropt  into  an  abyss,  should  be 
stopped  at  the  end  of  the  11th  second  after  its  delivery  4 what 
space  would  it  have  gone  through  ? Ans.  154,6TV  feet, 

QyEST.  42.  What  is  the  difference  between  the  depths  of 
two  wells,  into  each  of  which  should  a stone  be  dropped  at 
the  same  instant,  the  one  will  strike  the  bottom  at  6 seconds 
the  other  at  10  ? Ans.  1029ifeet. 

Quest.  43.  If  a stone  be  19^  seconds  in  descending  from 
the  top  of  a precipice  to  the  bottom,  what  is  its  height  ? 

Ans.  6115-j-i  feet. 

Quest.  44.  In  what  time  will  a musket  ball,  dropped  from 
the  top  of  Salisbury  steeple,  said  to  be  400  feet  high,  reach 
the  bottom  ? Ans.  5 seconds  nearly. 

Quest.  45.  If  a heavy  body  be  observed  to  fall  through 
100  feet  in  the  last  second  of  time,  from  what  height  did  it  fall, 
and  how  long  was  it  in  motion  ? 

Ans.  time  3§f£  sec.  and  height  209  feet. 

Quest.  46.  A stone  being  let  fall  into  a well,  it  was  observ- 
ed that,  after  being  dropped,  it  was  10  seconds  before  the 
sound  of  the  fall  at  the  bottom  reached  the  ear.  What  is  the 
depth  of  the  well  ? Ans.  1270  feet  nearly. 

Ql’ESt.  47.  It  is  proposed  to  determine  the  length  of  a 
pendulum  vibrating  seconds,  in  the  latitude  of  London, 
where  a heavy  body  falls  through  16TLfeet  in  the  first  second 
of  time  1 Ans.  39T1  inches. 

By  experiment  this  length  is  found  to  be  39£  inches. 

Quest.  48. 


254 


PRACTICAL  EXERCISEb 


Quest.  48.  What  is  the  length  of  a pendulum  vibrating  in 
2 seconds  : also  in  half  a second  and  in  a quarter  second  ? 

Ans.  the  2 second  pendulum  '56^ 
the  J second  pendulum  9|4 

the  second  pendulum  2t|T  inches. 

Quest.  49.  What  difference  will  there  be  in  the  number  of 
vibrations,  made  by  a pendulum  of  6 inches  long,  and  another 
of  12  inches  loDg,  in  an  hour’s  time  ? Ans.  26921. 

Quest.  50.  Observed  that  while  a stone  was  descending, 
to  measure  the  depth  of  a well,  a string  and  plummet,  that 
from  the  point  of  suspension,  or  the  place  where  it  was  held, 
to  the  centre  of  oscillation,  measured  just  ’S  inches,  had  made 
8 vibrations,  when  the  sound  from  the  bottom  returned.  Vt  bat 
was  the  depth  of  the  well  ? Ans.  412-Cl  feet. 

Quest.  51.  If  a hall  vibrate  in  the  arch  of  a circle,  10  de- 
grees on  each  side  of  the  perpendicular  ; or  a ball  roll  down  ■ 
the  lowest  10  degrees  of  the  arch  ; required  the  velocity  at 
the  lowest  point  ? the  radius  of  the  circle,  or  length  of  the  ; 
pendulum,  being  20  feet.  Ans.  4 4213  feet  per  second.  , 

Quest.  52.  If  a ball  descend  down  a smooth  inclined 
plane,  whose  length  is  100  feet,  and  altitude  10  feet  ; how 
long  will  it  be  in  descending,  and  what  will  be  the  last  ve-  - 
locity  ? 

Ans.  the  veloc.  25-364  feet  per  sec.  and  time  7 8852  sec. 

Quest.  53.  If  a cannon  hall,  of  lib  weight,  be  fired  against 
a pendulous  block  of  wood,  and  striking  the  centre  cl  oscilla- 
tion, cause  it  to  vibrate  an  arc  whose  chord  is  30  inches  ; the 
radius  of  that  arc,  or  distance  from  the  axis  tc  the  lowest  point 
of  the  pendulum  being  118  inches,  and  the  pendulum  vibra'ing 
in  small  arcs  40  oscillations  per  minute.  Required  the  velo- 
city of  the  ball,  and  the  velocity  of  the  centre  of  oscillation 
of  the  pendulum,  at  the  lowest  point  of  the  arc  ; the  whole 
weight  of  the  pendulum  being  5001b  ? 

Ans.  veloc  ball  1956-6054  feet  per  sec. 
and  veloc.  cent,  oscil.  3-9054  feet  per  sec. 

Quest.  54.  How  deep  will  a cube  of  oak  sink  in  ccmmou 
water  ; each  side  of  the  cube  being  1 foot  ? 

* Ans.  11TV  inches. 

Quest.  55.  How  deep  will  a globe  of  oak  sink  in  water  ; 
the  diameter  being  1 foot  ? Ans.  9-9867  inches. 

Quest. 


IN  NATURAL  PHILOSOPHY. 


255 


Quest.  56.  if  3 rube  of  wood,  floating  in  common  water, 
have  three  inches  of  if  dry  above  the  water,  and  4T|5  inches 
dry  when  in  sea  water  ; it  i?  proposed  *o  determine  the 
magnitude  of  the  cube,  and  what  •*  of  w -od  it  is  made  of  ? 

Ans.  the  wood  is  calc,  and  each  side  40  inches'. 

Quest  57.  An  irregular  piece  of  lead  ore  weighs,  in  air 
12  ounces,  but  in  water  only  7 ; and  another  fragment 
weighs  in  air  I4i  ounces,  but  in  water  only  9 ; required  their 
comparative  densities,  or  specific  gravities  ? 

• Ans.  as  145  to  132. 

Quest.  58.  An  irregular  fragment  of  glass,  in  the  scale, 
weighs  171  grains,  and  another  of  magnet  102  grains;  but 
in  water  the  first  fetches  up  no  more  than  120  grains,  and 
the  other  79  : what  then  will  their  specific  gravities  turn  out 
to  be  ? Ans.  glass  to  magnet  as  3933  to  5202 

or  nearly  as  10  to  13. 

Quest.  59.  Hiero,  king  of  Sicily,  ordered  his  jeweller  to 
make  him  a crown,  containing  03  ounces  of  gold.  The 
workmen  thought  that  substituting  part  silver  was  only  a 
proper  perquisite  ; which  taking  air,  Archimedes  was  ap- 
pointed to  examine  it ; who  on  putting  it  into  a vessel  of 
water,  found  it  raised  the  fluid  8*2245  cubic  inches  : and 
having  discovered  that  the  inch  of  gold  more  critically 
weighed  10  36  ounces,  and  that  of  silver  but  5*85  ounces,  he 
found  by  calculation  what  part  of  the  king’s  gold  had  been 
changed.  And  you  are  desired  to  repeat  the  process. 

Ans.  28*8  ounces. 

Quest.  60.  Supposing  the  cubic  inch  of  common  glass 
weigh  1-4921  ounces  troy,  the  same  of  sea-water  *59542,  and 
of  brandy  *5368  ; then  a seaman  having  a galloD  of  this 
liquor  in  a glass  bottle,  which  weighs  3*.841b  out  of  water, 
and,  to  conceal  it  from  the  officers  of  the  customs,  throws 
it  overboard.  It  is  proposed  to  determine,  if  it  will  sink,  how 
much  force  will  just  buoy  it  up  ? 

Ans.  14*1496  ounces. 

Quest.  61.  Another  person  has  half  an  anker  of  brandy, 
of  the  same  specific  gravity  as  in  the  last  question  ; the  wood 
of  the  cask  suppose  measures  i of  a cubic  foot ; it  is  proposed 
to  assign  what  quantity  of  lead  is  just  requisite  to  keep  the 
cask  and  liquor  under  water  ? Ans.  89  743  ounces. 

Quest.  62.  Suppose,  by  measurement,  it  be  found  that  a 
man-of-war,  with  its  ordnance,  rigging,  and  appointments, 

sinks 


256 


PRACTICAL  EXERCISES 


sinks  so  deep  as  to  displace  50000  cubic  feet  of  fresh  watei 
what  is  the  whole  weight  of  the  vessel  ? 

Ans.  1395T\  tons 

Quest.  63.  It  is  required  to  determine  what  would  be  the 
height  of  the  atmosphere,  if  it  were  every  where  of  the 
same  density  as  at  the  surface  of  the  earth,  when  the  .quick- 
silver in  the.  barometer  stands  at  30  inches  ; and  also,  what 
would  be  the  height  of  a water  barometer  at  the  same  time  . 

Ans.  height  of  the  air  29166f  feet,  or  5-5240  miles., 
height  of  water  35  feet. 

Quest.  64.  With  what  velocity  would  each  of  those 
three  fluids,  viz.  quicksilver,  water,  and  air,  issue  through 
a small  orifice  in  the  bottom  of  vessels,  of  the  respective 
heights  of  30  inches,  35  feet,  and  5-5240  miles,  estimating 
the  pressure  by  the  whole  altitudes,  and  the  air  rushiug  into  a 
vacuum  ? 

Ans.  the  velec.  of  quicksilver  12-681  feet 
the  veloc.  of  water  - 47  447 

the  veloc.  of  air  - - 1369-8 

Quest.  65.  A very  large  vessel  of  10  feet  high  (no  matter 
what  shape)  being  kept  constantly  full  of  water,  by  a large 
supplying  cock  at  the  top  ; if  9 small  circular  holes,  each  i of 
an  inch  diameter,  be  opened  in  its  perpendicular  side  at  every 
foot  of  the  depth  : it  is  required  to  determine  the  several  dis- 
tances to  which  they  will  spout  on  the  horizontal  plane  of  the 
base,  and  the  quantity  of  water  discharged  by  all  of  them  ir. 
10  minutes  ; 


Ans.  the  distances  are 
y/  36  or  6-00000 
^ 64  - 8-00000 

y/  S4  - 9-16515 

y/  96  - 9-79796 

y/  100  - 10-00000 

y/  96  - 9-79796 

^84  - 9-16515 

64  - 8-80000 

y/  36  -.  6-00000 

and  the  quantity  discharged  in  10  min.  123  8849  gallons. 


Note..  In  this  solution,  the  velocity  of  the  water  is  supposed 
to  be  equal  to  that  which  is  acquired  by  a heavy  body  in  falling 
through  the  whole  height  of  the  water  above  the  orifice,  and 
that  it  is  the  same  in  every  part  of  the  holes. 


Quest. 


IN  NATURAL  PHILOSOPHY.  257 

I - . ' , , 

Quest.  66.  If  the  inner  axis  of  a hollow  globe  of  copper, 
exhausted  of  air,  be  100  feet  ; what  thickness  must  it  be  of, 
that  it  may  just  float  in  the  air  ? 

Ans.  -02688  of  an  inch  thick. 

Quest.  67.  If  a spherical  balloon  of  copper,  of  ta7  of  an 
inch  thick,  have  its  cavity  of  100  feet  diameter,  and  be  filled 
with  inflammable  air,  of  of  the  gravity  of  common  air, 
what  weight  will  just  balance  it,  and  prevent  it  from  rising  up 
into  the  atmosphere  ? Ans.  212731b. 

Quest.  68.  If  a glass  tube,  36  inches  long,  close  at  top  be 
sunk  perpendicularly  into  water,  till  its  lower  or  open  end  be 

a 30  inches  below  the  surface  of  the  water  ; how  high  will  the 
water  rise  within  the  tube,  the  quicksilver  in  the  common  ba- 
rometer at  the  same  time  standing  at  29i  inches  ? 

Ans.  2-26545  inches. 

I Quest.  69.  If  a diving  bell,  of  the  form  of  a parabolic 
conoid,  be  let  down  into  the  sea  to  the  several  depths  of 
5,  10,  15,  and  20  fathoms  ; it  is  required  to  assign  the  res- 
pective heights  to  which  the  water  will  rise  within  it : its  ax- 
is and  the  diameter  of  its  base  being  each  8 feet,  and  the 
quicksilver  in  the  barometer  standing  at  30*9  inches  ? 

Ans.  at  5 fathoms  deep  the  water  rises  2-03516  feet, 
at  10  - - - - 3-06393 

. at  15  - - - - 3-70267 

at  20  - *-  - 4-14653 


[ 258  ] 


ON  THE  NATURE  AND  SOLUTION  OF  EQUATIONS 
IN  GENERAL. 


1.  In  order  to  investigate  the  general  properties  of  the 
higher  equations,  let  there  be  assumed  between  an  unknown 
quantity  x,  and  given  quantities  a,  b,  c,  d,  an  equation  con- 
stituted of  the  continued  product  of  uniform  factors  : thus 
(x  — a)  X ( x-b ) X (x— c)  X (ar  — d)  = 0. 

This,  by  performing  the  multiplications,  and  arranging  the 
final  product  according  to  the  powers  or  dimensions  of  x, 
becomes 

x 4 x + abcd  — 0.  . . . (A) 


Now  it  is  obvious  that  the  assemblage  of  terms  which  compose 
the  first  side  of  this  equation  may  become  equal  to  nothing  in 
four  different  ways  ; namely,  by  supposing  either  x = a,  or  > 
x = b,  or  x — c,  or  x — d ; for  in  either  case  one  or  other 
of  the  factors  x—a,  x — b,  x — c,  x-d,  will  be  equal  to  no- 
thing, and  nothing  multiplied  by  any  quantity  whatever  will 
give  nothing  for  the  product  If  any  other  value  e be  put 
for  x,  then  none  of  the  factors  e — a,  e — b,  e — c,  e — d,  being 
equal  to  nothing,  their  continued  product  cannot  be  equal  to 
nothing.  There  are  therefore,  in  the  proposed  equation,  four 
roots  or  values  of  x ; and  that  which  characterizes  these  roots 
is,  that  on  substituting  each  of  them  successively  instead  of  x, 
the  aggregate  of  the  terms  of  the  equation  vanishes  by  the 
opposition  of  the  signs  -f-  and  — . 

The  preceding  equation  is  only  of  the  fourth  power  or  de- 
gree ; but  it  is  manifest  that  the  above  remark  applies  to 
equations  of  higher  or  lower  dimensions  : viz  that  in  general 
an  equation  of  any  degree  whatever  has  as  many  roots  as  there 
are  units  in  the  exponent  of  the  highest  power  of  the  un- 
known quantity,  and  that  each  root  has  the  property  of  ren- 
dering, by  its  substitution  in  place  of  the  unknown  quantity, 
the  aggregate  of  all  the  terms  of  the  equation  equal  to  no- 


It  must  be  observed  that  we  cannot  have  all  at  once  ±=a-  j 
x — b,  x = c.  &c.  for  the  roots  of  the  equation  ; but  that  the  1 
particular  equations  x — a = 0,  x — b = 0,  x — c — 0,  &c. 
obtain  only  in  a disjunctive  sense.  They  exist  as  factors  in  i 


thing. 


the 


EQUATIONS. 


259 


the  same  equation,  because  algebra  gives,  by  one  and  the  same 
formula,  not  only  the  solution  of  the  particular  problem  from 
which  that  formula  may  have  originated,  but  also  the  solution 
of  all  problems  which  have  similar  conditions.  The  differ- 
ent roots  of  the  equation  satisfy  the  respective  conditions  : 
and  those  roots  may  differ  from  one  another,  by  their  quanti- 
ty, and  by  their  mode  of  existence. 

It  is  true,  we  say  frequently  that  the  roots  of  an  equation 
are  x = a,  x = b,  x = c,  &c.  as  though  those  values  of  x 
existed  conjunctively  ; but  this  manner  of  speaking  is  an  ab- 
breviation, which  it  is  necessary  to  understand  in  the  sense 
explained  above. 

2.  In  the  equation  a all  the  roots  are  positive  ; but  if  the 
factors  which  constitute  the  equation  had  been  x + a,  x -f-  b, 
x -J-  c,  x -f-  d,  the  roots  would  have  been  negative  or  sub- 
tractive. Thus 

ri  ^ x3  'j  x2  -\-abc  ^ x -J-  abed  = 0. 

+ac  j + abd  I 

+ad  ( -\-acd  | 

+bc  ( 

+bd\ 

+cdj 

has  negative  roots,  those  roots  being  x = — a,  x = — b, 
x = — c,  x = — d : and  here  again  we  are  to  apply  them 
disjunctively. 

3.  Some  equations  have  their  roots  in  part  positive,  in  part 
negative.  Such  is  the  following  : 

x -f-  abc  — 0 (C) 


(B) 


-f -bed 


— a } x2  -\-ab  i 
—6  > — ac  > 

+c)  —be) 


Here  are  the  two  positive  roots,  viz  x — a,  x = b ; and  one 
negative  root,  viz.  x = — c : the  equation  being  constituted 
of  the  continued  product  of  the  three  factors,  x — a = 0,  x — b 
= 0,  x + c = 0. 

From  an  inspection  of  the  equations  a,  b,  c,  it  may  be  in- 
ferred, that  a complete  equation  consists  of  a number  of  terms 
exceeding  by  unity  the  number  of  its  roots. 

4.  The  preceding  equations  have  been  considered  as  form- 
ed from  equations  of  the  first  degree,  and  then  each  of  them 
contains  so  many  of  those  constituent  equations  as  there  are 
units  in  the  exponent  of  its  degree.  But  an  equation  which 
exceeds  the  second  dimension,  may  be  considered  as  composed 
of  one  or  more  equations  of  the  second  degree,  or  of  the 
third,  &c.  combined,  if  it  be  necessary,  with  equations  of  the 
first  degree,  in  such  manner,  that  the  product  of  all  those 
constituent  equations  shall  form  the  proposed  equation.  In- 
deed, 


2G0 


EQUATIONS. 


deed,  when  an  equation  is  fenced  by  the  successive  multipli- 
cation of  several  simple  equations,  quadratic  equations,  cubic 
equations,  &c.  are  formed  ; which  of  course  may  be  regarded 
as  factors  of  the  resulting  equation. 

5.  It  sometimes  happens  that  an  equation  contains  imagi- 
nary roots  ; and  then  they  will  be  found  also  in  its  consti- 
tuent equations  This  class  of  roots  always  enters  an  equation 
by  pairs  ; because  they  may  be  considered  as  containing,  in 
their  expression  at  least,  one  even  radical  place  before  a ne- 
gative quantity,  and  because  an  even  radical  is  necessarily 
preceded  by  the  double  sign  ±.  Let,  for  example,  the  equa- 
tion be  x 4 — (2 a — 2c):r3-[-  a2  -Rf)2  — 4oc-Rc2  -f  d2  )x2  + (2a2c-(- 
2 b2c — 2ac2- — 2ad2)x4-(a2  -f-  b2 ) . (c2  -R  d-)  =0.  This  may 
be  regarded  as  constituted  of  the  two  subjoined  quadratic 
equations,  x2  — 2 ax-Ra2  -R6 2 = 0,  x2  -R2cx  -R  c2  -f  d2=0  : 
and  each  of  these  quadratic  contains  two  imaginary  roots  ; 
the  first  giving  x = a ± b - — 1,  and  the  second  x — — c ±: 

V-  1- 

In  the  equation  resulting  from  the  product  of  these  two 
quadratics,  the  coefficients  of  the  powers  of  the  unknown 
quantity,  and  of  the  last  term  of  the  equation,  are  real  quan- 
tities, though  the  constituent  equations  contain  imaginary 
quantities  ; the  reason  is,  that  these  latter  disappear  by  means 
of  addition  and  multiplication. 

The  same  will  take  place  in  the  equation  (x  — «)  . (x  -j-  b)  . 
(,r2  -j-  2cx  -R  c2  -R  d2)  — 0,  which  is  formed  of  two  equations 
of  the  first  degree,  and  one  equation  of  the  second  whose 
roots  are  imaginary. 

These  remarks  being  premised,  the  subsequent  genera! 
theorems  will  be  easily  established. 

THEOREM  I. 

Whatever  be  the  Species  of  the  Roots  of  an  Equation,  when 
the  Equation  is  arranged  according  to  the  Rowers  of  the. 
Unknown  Quantity,  if  the  First  Term  be  positive,  and 
have  unity  for  its  Coefficient,  the  following  Properties  may 
be  traced  : 

I.  The  first  term  of  the  equation  is  the  unknown  quantity 
raised  to  the  power  denoted  by  the  number  of  roots. 

II  The  second  term  contains  the  unknown  quantity  raised 
to  a power  less  than  the  former  by  unity,  with  a coefficient 
equal  to  the  sum  of  the  roots  taken  with  contrary  signs. 

III.  The  third  term  contains  the  unknown  quantity  raised 
to  a power  less  by  2 than  that  of  the  first  term,  with  a coeffi- 
cient equal  to  the  sum  of  all  the  products  which  can  be  form- 
ed by  multiplying  all  the  roots  two  and  two. 


EQUATIONS, 


261 


IV.  The  fourth  term  contains  the  unknown  quantity  raised 
to  a power  less  by  3 than  that  of  the  first  term  with  a coeffi- 
cient equal  to  the  sum  of  all  the  products  which  can  be  made 
by  multiplying  any  three  of  the  roots  with  contrary  signs. 

V.  And  so  on  to  the  last  term,  which  is  the  continued  pro- 
duct of  all  the  roots  taken  with  contrary  signs. 

All  this  is  evident  from  inspection  of  the  equations  exhibit- 
ed in  arts  1,  2,  3,  5. 

Cor.  1.  Therefore  an  equation  having  all  its  roots  real, 
but  some  positive,  the  others  negative,  will  want  its  second 
term  when  the  sum  of  the  positive  roots  is  equal  to  the  sum 
of  the  negative  roots.  Thus,  for  example,  the  equation  c 
will  want  its  second  term,  if  a -}-  b — c. 

Cor.  2.  An  equation  whose  roots  are  all  imaginary,  will 
want  the  second  term,  if  the  sum  of  the  real  quantities  which 
enter  into  the  expression  of  the  roots,  is  partly  positive,  partly 
negative,  and  has  the  result  reduced  to  nothing,  the  imagina- 
ry parts  mutually  destroying  each  other  by  addition  in  each 
pair  of  roots.  Thus,  the  first  equation  of  art.  5 will  want 
the  second  term  if  — ■ 2a  -f-  2c  = 0,  or  a — c.  The  second 
equation  of  the  same  article,  which  has  its  roots  partly  real, 
partly  imaginary,  will  want  the  second  term  if  b — a -j-  2c  = 
0,  or  a — 6 = 2c. 

Cor.  3.  An  equation  will  want  its  third  term,  if  the  sum 
of  the  products  of  the  roots  taken  two  and  two,  is  partly  po-. 
I sitive,  partly  negative,  and  these  mutually  destroy  each  other. 

Remark.  An  incomplete  equation  may  be  thrown  into  the 
form  of  complete  equations,  by  introducing,  with  the  coefficient 
a cypher,  the  absent  powers  of  the  unknown  quantity  : thus, 
for  the  equation  x3  + r — 0,  may  be  written  x3  -j-  0 x2  -f-  0 
x + r — 0.  This  in  some  cases  will  be  useful. 

Cor.  4.  An  equation  with  positive  roots  may  be  trans- 
formed into  another  which  shall  have  negative  roots  of  the 
j same  value,  and  reciprocally.  In  order  to  this,  it  is  only  ne- 
cessary to  change  the  signs  of  the  alternate  terms,  beginning 
i with  the  second.  Thus,  for  example,  if  instead  of  the  equa- 
tion x3  — 8x2  -j-  17a;—  10  = 0,  which  has  three  positive  roots 
1,  2,  and  5,  we  write  x3  -J-  8a:2  -f-  17a;  + 10  — 0,  this  latter 
equation  will  have  three-negative  roots  a;  = — 1,  x — — 2, 
x = — 5.  In  like  manner,  if  instead  of  the  equation  x3  + 

' 2a;2  — 13a:-}-  10^0,  which  has  two  positive  roots  x = l,x  — 2, 

1 and  one  negative  root  x = — 5,  there  be  taken  x 3 — 2a;2  — 
^3x  — 10  = 0,  this  latter  equation  will  have  two  negative 
roots,  x — — 1,  x — — 2,  and  one  positive  rootx  = 5. 

In  general,  if  there  be  taken  the  two  equations,  (a;  — a)  X 
(x  — b)  X (x— c)  X (x—d)  X &c  = 0,  and  fx-}-a)  X(ar+6)  X 


262 


equations. 


(x-f-c)  X(-r-j-d)  X &.c.  = 0,  of  which  the  roots  are  the  same 
in  magnitude,  but  with  different  signs  : if  these  equations  be 
developed  by  actual  multiplication,  and  the  term>  arranged 
according  to  the  powers  of  x,  as  in  arts.  1,2;  it  will  be  seen 
that  the  second  terms  of  the  two  equations  will  be  affected 
with  different  signs,  the  third  terms  with  like  signs,  the  fourth 
terms  with  different  signs,  kc. 

When  an  equation  has  not  all  its  terms,  the  deficient  terms 
must  be  supplied  by  cyphers,  before  the  preceding  rule  can  be 
applied. 

Cor.  5.  The  sum  of  the  roots  of  an  equation,  the  sum  of 
their  squares,  the  sum  of  their  cubes,  &c.  may  be  found  with- 
out knowing  the  roots  themselves.  For,  let  an  equation  of 
any  degree  or  dimension,  m,  be  xm  -f-  fxm~x  + gxm~2  -f- 
hxm~ 3 + &c.  = 0,  its  roots  being  a,  b,  c,  d,  kc.  Then  , we 
shall  have, 

1st.  The  sum  of  the  first  powers  of  the  roots,  that  is,  of 
the  roots  themselves,  or  a -f  b + c + &c.  = — / ; since  the 
coefficient  of  the  unknown  quantity  in  the  second  term,  is 
equal  to  the  sum  of  the  roots  taken  with  different  signs. 

2dly.  The  sum  of  the  squares  of  the  roots,  is  equal  to 
the  square  of  the  coefficient  of  the  second  term  made  less  by 
twice  the  coefficient  of  the  third  term  : viz.  a-  -f  b2  + c2  + 
kc.  = /2  = 2g.  For,  if  the  polynomial  a + b -j-  c + &c.  be 
squared,  it  will  be  found  that  the  square  contains  the  sum  of 
the  squares  of  the  terms,  a,  b,  c,  kc.  plus  twice  the  sum  of 
the  products  formed  by  multiplying  two  and  two  all  the  roots 
a,  b,  c,  &c.  That  is,  (a+6-f  c+&c.)2  = a2  -j-  b2+c2  + kc. 
H-2(«6  + ac  + be  + &c.).  But  it  is  obvious,  from  equa.  a,  b, 
that  (a+b+c  -f  &c.)2  = /',  and  (ab  + ac  + be  + kc.)  =g, 
Thus  we  havejf2  = (a2  + b2  -f  c2-f  Lc.)  + 2 g : and  conse- 
quently a2  -j-  b-  + c2  -j-  &c.  z=f2  — 2 g. 

3dly.  The  sum  of  the  cubes  of  the  roots,  is  equal  to  3 times 
the  rectangle  of  the  coefficient  of  the  second  and  third  terms 
made  less  by  the  cube  of  the  coefficient  of  the  second  term, 
and  3 times  the  coefficient,  of  the  fourth  term  : viz.  a3  + b3 
_j_  c3  _j_  &c.  = — /3  + 3fg  — 3 h.  For  we  shall  by  actual  in- 
volution have  (a  + b 4-c-f&c.)3  = a.3  + b3  -f*  c3  + &.c.  + 
3(a-}-&+c)  X {ab  + be  + ac)  — 3 abc.  But  (a-f  i-j-c+k;c)a 

— f3,  (a  + b -(-  c + &c.)  X {ab  -fac-fic  +&c.)  = — fg, 
abc  =—  h‘  Hence  therefore,—  f3  — a3+b3  c3  + &c. — 
3j~g-\-3h  ; and  consequently,  a3  -f  b3  -f  c3  + kc  = — f3  + 

3 ftr  _ 3h.  And  so  on,  for  other  powers  of  the  roots. 

JO  THEOREM 


EQUATIONS. 


263 


THEOREM  IT. 

{ 

In  Every  Equation,  which  contain  only  Real  Roots  : 

I.  If  all  the  roots  are  positive,  the  terms  of  the  equation 
will  be  + and  — alternately. 

II.  If  all  the  roots  are  negative,  all  the  terms  will  have  the 
sign  -f-. 

III.  If  the  roots  are  partly  positive,  partly  negative,  there 
will  be  as  many  positive  roots  as  there  are  variations  of  signs, 
and  as  many  negative  roots  as  there  are  permanencies  of  signs  ; 
these  variations  and  permanences  being  observed  from  one 
term  to  the  following  through  the  whole  extent  of  the  equa- 
tion. 

In  all  these,  either  the  equations  are  complete  in  their  terms, 
or  they  are  made  so. 

The  first  part  of  this  theorem  is  evident  from  the  examina- 
tion of  equation  a ; and  the  second  from  equation  b. 

To  demonstrate  the  third,  we  revert  to  the  equation  c 
(art.  3),  which  has  two  positive  roots,  and  one  negative.  It 
may  happen  that  either  c > a-\-b , or  c 

In  the  first  case,  the  second  term  is  positive,  and  the  third 
is  negative  ; because,  having  c > a b,  we  shall  have  ac  -j- 
bc  > (a  + b)2  > ab.  And,  as  the  last  term  is  positive,  we  see 
that  from  the  first  to  the  second  there  is  a permanence  of 
signs  ; from  the  second  to  the  third  a variation  of  signs  ; and 
from  the  third  to  the  fourth  another  variation  of  signs.  Thus 
there  are  two  variations  and  one  permanence  of  signs  ; that 
is,  as  many  variations  as  there  are  positive  roots,  and  as  many 
permanences  as  there  are  negative  roots. 

In  the  second  case,  the  second  term  of  the  equation  is  ne- 
gative, and  the  third  may  be  either  positive  or  negative.  If 
that  term  is  positive,  there  will  be  from  the  first  to  the  second 
a variation  of  signs  ; from  the  second  to  the  third  another 
variation  ; from  the  third  to  the  fourth  a permanence  ; making 
in  all  two  variations  and  one  permanence  of  signs.  If  the 
third  term  be  negative  ; there  will  be  one  variation  of  signs 
from  the  first  to  the  second  ; one  permanence  from  the  second 
to  the  third  ; and  one  variation  from  the  third  to  the  fourth  : 
thus  making  again  two  variations  and  one  permanence.  The 
number  of  variations  of  signs  therefore  in  this  case,  as  well  as 
in  the  former,  is  the  same  as  that  of  the  positive  roots  ; and 
the  number  of  permanencies,  the  same  as  that  of  the  negative 
roots. 

Corol.  Whence  it  follows,  that  if  it  be  known,  by  any 
means  whatever,  that,  an  equation  contains  onlv  real  roots,  it 

is 


264 


EQUATIONS. 


is  also  known  how  many  of  them  are  positive,  and  how  manv 
negative.  Suppose,  for  example,  it  be  known  that,  in  the 
equation  x 5 -f  3x4  — 23x3  — 27x2  + 166x  — 120  = 0,  all 
the  roots  are  real  : it  may  immediately  be  concluded  that  there 
are  three  positive  and  two  negative  roots.  In  fact  this  equa- 
tion has  the  three  positive  roots  x = 1 , x — 2,  x = 3 ; and 
two  negative  roots,  x — — 4,  x = — 5. 

If  the  equation  were  incomplete,  the  absent  terms  must  be 
supplied  by  adopting  cyphers  for  coefficients,  and  those  terms 
must  be  marked  with  the  ambiguous  sign  ±.  Thus,  if  the 
equation  were 

xs—20x3  + 30x2  + 19x-30  = 0, 
all  the  roots  being  real,  and  the  second  term  wanting.  It 
must  be  written  thus  : 

-f-  Orr«  — 20x3  •—  30x2  -f  19xr — 30  = 0. 

Then  it  will  be  seen,  that,  whether  the  second  term  be  posi- 
tive or  negative,  there  will  be  3 variations  and  2 permanencies 
of  signs  : and  consequently  the  equation  has  3 positive  and  2 
negative  roots.  The  roots  in  fact  are,  1,  2,  3,  — 1,  — 5. 

This  rule  only  obtains  with  regard  to  equations  whose  roots 
are  real.  If,  for  example,  it  were  inferred  that,  because  the 
equation  x2  + 2x  -j-  5 = 0 had  two  permanencies  of  signs,  it 
had  two  negative  roots,  the  conclusion  would  be  erroneous 
for  both  the  roots  of  this  equation  are  imaginary. 

THEOREM  IH. 

Every  Equation  may  be  Transformed  into  Another  whose 
Roots  shall  be  Greater  or  Less  by  a Given  Quantity. 

In  any  equation  whatever,  of  which  x is  unknown,  (the 
equations  a,  b,  c,  for  example)  make  x = z + m,  z being  a 
new  unknown'  quantity,  m any  given  quantity,  positive  or 
negative  ~ then  substituting,  instead  of  x and  its  powers,  their 
values  resulting  from  the  hypothesis  that  .r  = z + m ; so  shall 
there  arise  an  equation,  whose  roots  shall  be  greater  or  less 
than  the  roots  of  the  primitive  equation,  by  the  assumed 
quantity  m. 

Carol.  The  principal  use  of  this  transformation,  is,  to  take 
away'  any  term  out  of  an  equation.  Thus,  to  transform  an 
equation  into  one  which  shall  want  the  second  term,  let  m be 

so  assumed  that  nm—a  = 0,  or  m = -,  n being  the  index  of 

77 

the  .highest  power  of  the  unknown  quantity,  and  a the  coeffi- 
cient of  the  second  term  of  the  equation,  with  its  sign  changed  : 
then,  if  the  roots  of  the  transformed  equation  can  be  found, 
the  roots  of  the  original  equation  may  also  be  found,  be- 

i " 

cause  x = z - f-  -. 

n 


THEOREM 


EQUATIONS. 


265 


THEOREM  IV. 


Every  Equation  may  be  Transformed  into  Another,  whose 
Roots' shall  be  Equal  to  the  hoots  of  the  First  Multiplied 
or  Divided  by  a Given  Quantity. 


1.  Let  the  equation  be  z 3 + az 2 + bz  -f-  c = 0 : if  we  put 
fz  — x,  or  z = j.,  the  transformed  equation  will  be  x3  -f- 

fax24-f2bx-{-f3c  ~ 0,  of  which  the  roots  are  the  respective 
products  of  the  roots  of  the  primitive  equation  multiplied  into 
the  quantity  f. 

By  means  of  this  transformation,  an  equation  with  frac- 
tional  quantities,  may  be  changed  into  another  which  shall 

• CLZ  ^ 

be  free  from  them.  Suppose  the  equation  were  z3-j— — -f- 
bz  d • 

— + -=  0 • multiplying  the  whole  by  the  product  of  the 


denominators,  there  would  arise  ghkz3  + hkaz2  -f-  gkbz  -f- 
ghd  = 0 : then  assuming  ghkz  = x,  or  z = the  transform- 
ed equa.  would  be  x3  -f-  hkax2  -^-g2k2hbx-\-g3k3h3d  = 0. 

The  same  transformation  may  be  adopted,  to  exterminate 
the  radical  quantities  which  affect  certain  terms  of  an  equa- 
tion. Thus,  let  there  be  given  the  equation  z3  4-  az2  y k -f* 
bz  -f-  c y k : make  z^/k  — x ; then  will  the  transformed 
equation  be  x3  + akx2-\-bkx  + ck2  = 0,  in  which  there  are 
no  radical  quantities. 

2.  Take,  for  one  more  example,  the  equation  z3  -j-  az2  -f- 
bz  -f-  c = 0.  Make  j.  — x ■ then  will  the  equation  be 
dx^  bx  c 

transformed  to  x3  + — j— — + = 0,  in  which  the  roots 

j ./ 2 J6 

are  equal  to  the  quotients  of  those  of  the  primitive  equations 
divided  by/. 

It  is  obvious  that,  by  analogous  methods,  an  equation  may 
be  transformed  into  another,  the  roots  of  which  shall  be  to 
those  of  the  proposed  equation,  in  any  required  ratio.  But 
.he  subject  need  not  be  enlarged  on  here.  The  preceding 
succinct  view  will  suffice  for  the  usual  purposes,  so  far  as  re- 
lates to  the  nature  and  chief  properties  of  equations.  We  shall 
herefore  conclude  this  chapter  with  a summary  of  the  most 
iseful  rules  for  the  solution  of  equations  of  different  degrees, 
resides  those  already  given  in  the  first  volume 


Vor,.  II. 


35 


1.  Rules 


266  SOLUTION  OF  EQUATIONS  BY  SINES  &c. 


I.  Rules  jor  the  Solution  of  Quadratics  by  Tables  of  Sines  and 
Tangents. 


1.  If  the  equation  be  of  the  form  x2  px  = q : 

2 

Make  tan  a = -^/  q ; then  will  the  two  roots  be, 

P 

x = -f-  tan  q x — — cot  aa y/q. 

2.  For  quadratics  of  the  form  xs  — px  = q. 

2 

Make,  as  before,  tan  a — -\/q  : then  will 

x = — tan  |a y/q x = + cot  \ky/q. 

3.  For  quadratics  of  the  form  x2  -f-  px  — — q. 

. 2 

Make  sin  a = -y/q  : then  will 
P 

x — — tan  1a ^/q x = — cot  %Ay/q- 

4.  For  quadratics  of  the  form  x2  — px  = — q. 

2 

Make  sin  a = -y/q  : then  will 
PV^ 

x = + tan  \Ay/q x — + cot  {a y/q. 

2 

In  the  last  two  cases,  ^ -■/  9 exceed  unity,  sin  a is  imagi- 
nary, and  consequently  the  values  of  x. 

The  logarithmie  application  of  these  formulae  is  very  sim- 
ple. Thus,  in  case  1st.  Find  a by  making 

10  -j-  log  2 -f- 1 log  q — log  p = log  tan  a. 

Then  Io2  x = $ + lo2 tan  + i lo§  10\ 

S ( - (l°g  cot  +1  log  q-  10). 

Note.  This  method  of  solving  quadratics,  is  chiefly  of  use 
when  the  quantities  p and  q are  large  integers,  or  complex 
fractions. 


II.  Rules  for  the  Solution  of  Cubic  Equation  by  tables  of  Sines-. 
Tangents , and  Secants. 


1.  For  cubics  of  the  form  x3  4 -px  ±q  — 0. 

Make  tan  b =t£.  2V»%> tan  a = %/  tan  Ab- 

? Then  x — =F  cot  2a  . 2y/\ p. 

2.  For  cubics  of  the  for.m  x3  — px  ± <7  = 0. 

Make  sin  b = — . 2 y/\p tan  a = ^/tan  £b. 

q 

Then  =p  cosec  2a  . 2 y/\p. 

Here,  if  the  value  of  sin  b should  exceed  unity,  b would  be 
imaginary,  and  the  equation  wovld  tall  in  what  is  called 

the 


SOLUTION  OF  EQUATIONS  BY  SINES  &c.  2.67 


the  irreducible  case  of  cubics.  In  that  case  we  must  make 

cosec  3a  = — . V ip  : and  then  the  three  roots  would  be 

x = ± sin  a . 2 y/  \p. 
x = ± sin  (60° — a)  . 2i/±p. 
x — ± sin  (60° -{-a)  • 

If  the  value  of  sin  b were  1,  we  should  have  b = 90°,  tan 
a=  1 ; therefore  a = 45°,  and  i = + ip.  But  this 
would  not  be  the  only  root.  The  second  solution  would  give 

90° 

cosec  3a  = 1 ; therefore  a = — ; and  then 

O 

x = it  sin  30°  . 2^/i p = ± v/iP' 
x — sin  30°  . 2^/ip  = dz  ip. 
x = qz  sin  90°  . 2^/ip  = ^2^/i p. 

Here  it  is  obvious  that  the  first  two  roots  are  equal,  that  their 
sum  is  equal  to  the  third  with  a contrary  sign,  and  that  this 
third  is  the  one  which  is  produced  from  the  first  solution*. 

In  these  solutions,  the  double  signs  in  the  value  of  x,  re- 
late to  the  double  signs  in  the  value  of  q. 

N.  B.  Cardan’s  rule  for  the  solution  of  Cubics  is  given  in 
i the  first  volume  of  this  course. 


* The  tables  of  sines,  tangents,  &c.  besides  their  use  in  trigonometry, 
and  in  the  solution  of  the  equations,  are  also  very  useful  in  finding  the 
value  of  algebraic  expressions  where  extraction  of  roots  would  be 
otherwise  required.  Thus,  if  a and  b be  any  two  quantities,  of  which 

b b 

!a  is  the  greater.  Find  »,  z,  8tc.  so,  that  tan  x — , sin  z=  a/~  ,sec 

a a 

a b b 

y = — , tan  u——  and  sin  t — — : then  will 
b a a 

1°.?  s/(a2  — bz  ) — log  a + log  sin  y = log  b + log  tan  y. 
log  v'(a2_62)=4Q0g(a.4_£)  + log  (a— 6)]. 
log  v/(a2  -f-  62  ) =log  a + log  sec  u = log  b -f  log  cosec  u. 
log  v/(a+6)  =i  log  a + log  sec  x = Jiog  a+  J!og2  -f  log  cos  \y, 
log  -vAa — b)=h  log  a -f  log  cos  z — ^loga  -1-  4log2  -f-  log  sin  $y. 
m m 

log  (a±;6)  n = — [log  a + log  cos  t -f.  log  tan  45°  ± £{)]. 
n 

The  first  three  of  these  formulas  will  often  be  useful,  when  two  sides 
of  a right-angled  triangle  are  given,  to  find  the  third. 


III.  Solution 


268  SOLUTION  OF  BIQUADRATIC  EQUATIONS. 


III.  Solution  of  Biquadratic  Equations. 

Let  the  proposed  biquadratic  be  x 4 + 2 px3  — qx 2 -f - rx  + s. 
Now(x2  -f-px+t*)2  —x*  +2p.t3+  (p2  +2n)  x2 -\-2pnx-\-n2  : if 
therefore  (p2  + 2w)  x2  + 2 pnx  + w2  be  added  to  both  sides 
of  the  proposed  biquadratic,  the  first  will  become  a complete 
square  (x2  -f-  px  + «)2,  and  the  latter  part  (p2  +2 n q)  x2 
•+  Czpn  -f  r)  x + n2  -f  s,  is  a complete  square  if  4(p2  + 2 n 
+ 9)  • (n2  +s)  = (2p»  -j-  r)2 ; that  is,  multiplying  and  arranging 
the  terms  according  to  the  dimensions  of  n , if  8n3  -f-  49 n 2 + 
(8s  — 4rp)n  -j-  49s  -f-  4p2s  — r2  = 0.  From  this  equation 
let  a value  of  n be  obtained,  and  substituted  in  the  equation 
(x2  -f-  px  + n)2  — (p2  +2 n q)x2  + (2pn+x  ) x -f-  n2  +s  ; 
then  extracting  the  square  root  on  both  sides. 

x2+px-t-n=±  | v/(p2+2n-f  9)x+v/(n2+s) 

or x2+px+n  = ± J v/(p2+2n  + 9)x-v/(n2+s)  J 

And  from  these  two  quadratics,  the  four  roots  of  the  given 
biquadratic  may  be  determined*. 

Note.  Whenever,  by  taking  away  the  second  term  cf  a 
biquadratic,  after  the  manner  described  in  cor.  th.  3,  the 
fourth  term  also  vanishes,  the  roots  may  immediately  be  ob- 
tained by  the  solution  of  a quadratic  only. 

A biquadratic  may  also  be  solved  independently  of  cubics,  ; 
in  the  following  cases  : 

1.  When  the  difference  between  the  coefficient  of  the 
third  term,  and  the  square  of  half  that  of  the  second  term,  is 
equal  to  the  coefficient  of  the  fourth  term,  divided  by  half 
that  of  the  second.  1 hen  if  p be  the  coefficient  of  the  second 
term,  the  equation  will  be  reduced  to  a quadratic  bv  dividing 
it  by  x 2 ± \px. 

2.  When  the  last  term  is  negative,  and  equal  to  the  square 
of  the  coefficient  of  the  fourth  .term  divided  by  4 times  that 
of  the  third  term,  minus  the  square  of  that  of  the  second  : 
then  to  complete  the  square,  subtract  the  terms  of  the  pro- 
posed biquadiatic  from  (x2  ± ipx)2,  and  add  the  remainder 
to  both  its  sides. 

3 When  the  coefficient  of  the  fourth  term  divided  by 
that  of  the  second  term,  gives  for  a quotient  the  square  root 
of  the  last  term  : then  to  complete  the  square,  add  the  square 
of  half  the  coefficient  of  the  second  term,  to  twice  the  square 


wben2p7i-}-r 
is  positive  ; 


* 1 his  rule,  for  solving  biquadratics,  by  conceiving  each  to  be  the 
difference  of  two  squares,  is  frequently  ascribed  to  Do  Waring  ; but 
its  original  inventor  was  Mr.  Thomas  Simpson,  formerly  Professor  of 
Mathematics  in  the  Royal  Military’  Academy 


root 


EULER’S  RULE  FOR  BIQUADRATICS. 


26y 


root  of  the  last  term,  multiply  the  sum  by  x3 , from  the  pro- 
duct take  the  third  term,  and  add  the  remainder  to  both  sides 
of  the  biquadratics. 

4.  The  fourth  term  will  be  made  to  go  out  by  the  usual  op- 
eration for  taking  away  the  second  term,  when  the  difference 
between  the  cube  of  half  the  coefficient  of  the  second  term 
and  half  the  product  of  the  coefficients  of  the  second  and  third 
term,  is  equal  to  the  coefficient  of  the  fourth  term. 

IV.  Euler's  Rule  for  the  Solution  of  Biquadratics. 


Let  x4  — ax 2 — bx  — c = 0,  be  the  given  biquadratic  equa- 
tion wanting  the  second  term..  Take/  = ±a,  g — r\a3  -f-  ^c, 
and  h — , or  h = \b  : with  which  values  of  /,  g,  h, 

form  the  cubic  equation,  z3  — fz3  -f-  gz  — h = 0.  Find  the 
roots  of  this  cubic  equation,  and  let  them  be  called  p,  q,  r. 
Then  shall  the  four  roots  of  the  proposed  biquadratic  be  these 
following  : viz. 


When  }b  is  positive. 

1.  X = v^  + v^  + aA- 

2 x = y/p  — ]yq  — \/r . 
3.  x = — ^/p  + l/q~  \/r. 
4 x =-[/p—  l/q+i/r. 


When  \b  is  negative  : 
x — x/p+ y/9~\/r- 
X = A/P-V/?+A/r- 

x = — -v/P+a/9+aA- 
x ^ -v/P~\/(!—Vr- 


Note  1.  In  any  biquadratic  equation  having  all  its  terms, 
if  | of  the  square  of  the  coefficient  of  the  2d  term  be  greater 
than  the  product  of  the  coefficients  of  the  1st  and  3d  terms, 
or  f of  the  square  of  the  coefficient  of  the  4th  term  be  greater 
than  the  product  of  the  coefficients  of  the  3d  and  fifth  terms, 
»r  f of  the  square  of  the  coefficient  of  the  3d  term  greater 
than  the  product  of  the  coefficients  of  the  2d  and  4th  terms  ; 
then  all  the  roots  of  that  equation  will  be  real  and  unequal  ; 
but  if  either  of  the  said  parts  of  those  squares  be  less  than 
either  of  those  products,  the  equation  will  have  imaginary 
roots. 

2.  In  a biquadratic  x4  -f-  ox3  -f-  bx3  -f-  cx-f-  cZ=0,  of 
which  two  roots  are  impossible,  and  d an  affirmative  quantity, 
then  the  two  possible  roots  will  be  both  negative,  or  both  affir- 
mative, according  as  a3  — 4 ab  + 8c,  is  an  affirmative  or  a 
negative  quantity,  if  the  signs  of  the  coefficients  a,  b,  c,  d,  are 
neither  all  affirmative,  nor  alternately  — and  +*. 


* Various  general  rules  for  the  solution  of  equations  have  been  giv- 
by  Demoivre,  Bezout  Lagrange,  &c.  ; but  the  most  universal  in 
their  application  are  approximating  rules,  of  which  a very  simple  and 
useful  one  is  given  in  our  first  volume. 


EXAMPLES. 


270  NUMERAL  SOLUTIONS  OF  EQUATIONS. 


EXAMPLES. 

Ex.  1.  Find  the  roots  of  the  equation  x2  4-  — x — 1655  . 

H 44  12716 

by  tables  of  sines  and  tangents. 

7 1695 


Here  p = 


44  ’ 


7 = 


12716 
88 

1st,  form.  Also  tan  a=  — «/  „ 

7 v 12716 

In  logarithms  thus  : 

Log.  1695  = 3-2291697 
Arith.  com.  log.  12716  = 5-8956495 
sum  — {—  1 0 = 19-1248192 
half  sum  = 9-5624096 
log  88  = 1-9444827 

Arith.  com.  log  7 = 9-1549020 


, and  the  equation  agrees  with  the 

1695  , , , 1695 

, and  ,r=tan  — 


12716 


sum  — 10  = log  tan  a = 10-6617943  = log  tan  77°42'31'f  ; 

log  tan  Aa  = 9-9061115  = log  tan  38051'15"a  ; 
log  q,  as  above  = 9-5624096 
sum — 10  = log  x — — 1-4685211  = log  -2941176. 

This  value  of  x,  viz.  -2941 176, is  nearly  equal  to  ~ . Tofind 

whether  that  is  the  exact  root,  take  the  arithmetical  compli- 
ment of  the  last  logarithm,  viz.  0 5314379,  and  consider  it  as 
the  logarithm  of  the  denominator  of  a fraction  whose  nume- 
rator is  unity;  thus  is  the  fraction  found  to  be  — exactly, 


and  this  )3  manifestly  equal  to 
the  equation,  it  is  equal  to 


17’ 

1 695 


As  to  the  other  root  of 
i-A.  — _ ^.9 

748' 


12716  17 

Ex.  2.  Find  the  roots  of  the  cubic  equation. 

-I — — = 0,  by  a table  of  sines. 

441  147  J 

Here  p = — A q — i£,  the  second  term  is  negative,  and 
441  7 147 

4p 2 >27 q2 : so  that  the  example  falls  under  the  irreducible  case. 

- _ 3*46^441  v 1 _4l4  1 

>A  — 147  403  X 403  403  ' 1612* 


Hence,  sin 


2^3-441 

The  three  values  of  x therefore,  are 
. 1612 


^ 1623 


SOLUTIONS  OF  EQUATIONS. 


27 1 


The  logarithmic  computation  is  subjoined. 

Log  1612  = 3-2073650 
Arith.  com.  log  1323  = 6-8764402 

sum  — 10  = 0 0858"52 

half  sum  = 0-0429026  const.  log. 

Arith.  com.  const,  log.  = 9-957097  4 
log  414  ...  = 2-6170003 
Arith.  com.  log.  403  . = 7-3946950 

log  sin  3a  ...  = 9-9687927  = log  sin  68°  32'  18"f 
Log  sin  a = 9-5891206 
const.  log  = 0-0429026 

1.  sum— 10  = log  x — — 1-6320232  = log  -4285714=log^. 

Log  sin  (60° -a)  = 9-781006~l 
const,  log  . . . . = 0-0429026 

2.  sum — 10  = log  x — — 1-8239087  =log-6666666=log|. 

Log  sin  (60*-!- a)  = 9-9966060 
const,  log  . . . . = 0-0429026 

3.  sum— 10  = log  — x = 0-0395086=logl-095238=log|f. 
So  that  the  three  roots  are  §,  and  — § f ; of  which  the  first 
two  are  together  equal  to  the  third  with  its  sign  changed,  as 
they  ought  to  be. 


Ex.  3.  Find  the  roots  of  the  biquadratic  xk  — 25x2 
60a;— 36  = 0,  by  Euler’s  Rule. 

Here  a = 25,  b = — 60,  and  c = 36  ; therefore 
, 25  625  . 769  , , 225 

/ = 2--«  = -iff  + 9=  w’aD<u=7- 
Consequently  the  cubic  equation  will  be 

, 25  769  225 

z3  — — z2  -{- z — — = 0. 

2 16  4 

The  three  roots  of  which  are 


Z=Z4  ~P’ 


, 25 

and  z — — — r 
4 


and  z = 4 = q, 

the  square  roots  of  these  are  ^/p  = f , ^/q  = 2 or  r =§ , 
Hence,  as  the  value  of  %b  is  negative,  the  four  roots  are 


1st.  x = 'f-K-f  = L 

2d.  x = | — A+f  = 2, 

3d.  a?  = - t+i+4-  = 3. 

4th.  x — — f — | — f = —6. 

Ex.  4.  Produce  a quadratic  equation  whose  roots  shall  be 
J and  A-  Ans.  a;2  — |a.t  -j-  ^ = 0. 

Ex.  5.  Produce  a cubic  equation  whose  roots  shall  be,  2,  5> 
and  — 3.  Ans.  a:3  — 4.v2  — 1 lx  + 30  = 0. 

Ex.  6. 


272 


SOLUTIONS  OF  EQUATIONS. 


Ex.  6.  Produce  a biquadratic  which  shall  have  for  the  roov 
1,  4,-5,  and  6 respectively. 

Ans.  x4  — 6a3  -21a2  -f-  146a-  120=  0. 
Ex.  7.  Find  x,  when  x2  -f-  347a  = 22110 

Ans.  x = 55,  x = — 402 

Ex.  8. 


Find  the  roots  of  the  quadratic  x 2 — x 


Ans.  x — 10,  x = — 


325 

TP 

65 

12 


Ex.  9. 


264 

Solve  the  equation  x2  - — x 
25 


695 

25* 


a , 159 

Ans.  x = 5,  x = — 
’ 25 


Ex.  10.  Given  x2—  2411 3x  — 481860,  to  find  x. 

Ans.  x = 20  x = 24095. 

Ex.  11.  Find  the  roots  of  the  equation  a3— 3a  — 1 = 0. 

Ans.  the  roots  are  sin  70°,  — sin  50°,  and  — sin  10°,  to  a 
radius  = 2 ; or  the  roots  are  twice  the  sines  of  those  arcs  as 
given  in  the  tables. 

Ex.  12.  Find  the  real  root  of  x3  —x—6  = 0. 

Ans.  f ^3  X sec  54°  44'  20' . 

Ex.  13.  Find  the  real  root  of  25a3  -j-  75a:—  46  = 0. 

Ans.  2 cot  74°  27'  48  '. 

Ex.  14.  Given  a4 — 8a3  — 12a2  -{- 84a— 63  = 0,  to  find  a 
by  quadratics.  Ans.  o' = 2 + 1+v'7- 

Ex.  15.  Given  a4-|-36a3  — 400a2  — 3168a  + 7744  = 0,  to 
find  x,  by  quadratics.  Ans.  x = 1 1 -{-  209, 

Ex.  16.  Given  a44"24a3 — 1 14a2  —24a-  + 1 = 0 to  find  x. 

Ans.  x = ± 197 — 14,  x — 2 ± 5 

Ex.  17.  Find  x,  when  x4  — 12a-— 5 = 0. 

Ans.  x = 1 ± y/2,  x — — 1 ± 2^/  — 1. 

Ex.  18.  Find  a,  when  a4  — 12a3  + 47a3  — 72a+ 36  = 0. 

Ans.  x = 1 , or  2,  or  3,  or  6. 

Ex.  19.  Given  a5  — 5aa4  — 80a3a3  — 68<z2a2 +7a4a4-5  = 0. 
to  find  a. 

Ans.  a — a,  a = 6a  ± a^/ 37,  x — zt  a 10—  3a. 


[ 273  j 


ON  THE  NATURE  AND  PROPERTIES  OF  CURVES,  AND  THE 
CONSTRUCTION  OF  EQUATIONS. 


SECTION  I. 

Nature  and  Properties  of  Curves. 

Def.  1.  A carve  is  a line  whose  several  parts  proceed  in 
different  directions,  and  are  successively  posited  towards  dif- 
ferent points  in  space,  which  also  may  be  cut  by  one  right 
line  in  two  or  more  points. 

If  all  the  points  in  the  curve  may  be  included  in  one  plane, 
the  curve  is  called  a plane  curve  ; but  if  they  cannot  all  be 
comprised  in  one  plane,  then  is  the  curve  one  of  double  cur- 
vature. 

Since  the  word  direction  implies  straight  lines,  and  in  strict- 
ness no  part  of  a curve  is  a right  line,  some  geometers  prefer 
defining  curves  otherwise  : thus,  in  a straight  line,  to  be  called 
the  line  of  the  abscissas,  from  a certain  point  let  a line  arbi- 
trarily taken  be  called  the  abscissa,  and  denoted  (commonly) 
by  x : at  the  several  points  corresponding  to  the  different 
values  of  x,  let  straight  lines  be  continually  drawn,  making  a 
i certain  angle  with  the  line  of  the  abscissas  : these  straight  lines 
being  regulated  in  length  according  to  a certain  law  or  equa- 
tion, are  called  ordinates  ; and  the  line  or  figure  in  which 
their  extremities  are  continually  found  is,  in  general,  a curve 
line.  This  definition  however  is  not  free  from  objection  ; 
for  a right  line  may  be  denoted  by  an  equation  between  its 
abscissas  and  ordinates,  such  as  y = ax  -f-  b. 

Curves  are  distinguished  into  algebraical  or  geometrical, 
and  transcendental  or  mechanical. 

Def.  2.  Algebraical  or  geometrical  curves,  are  those  in 
which  the  relations  of  the  abscissas  to  the  ordinates  can  be 
denoted  by  a common  algebraical  expression  ; such,  for  ex- 
ample, as  the  -equations  to  the  conic  sections,  given  in  page 
532  &c.  of  vol.  2. 

Def.  3.  Transcendental  or  mechanical  curves,  are  such  as 
cannot  be  so  defined  or  expressed  by  a pure  algebraical  equa- 
tion ; or  when  thev  are  expressed  by.  an  equation,  having  one 

Vor..  II  " 36  of 

• -- 

i j 


274  NATURE  AND  PROPERTIES  OF  CURVES. 


of  its  terms  a variable  quantity,  or  a curve  line.  Thus,  y = 
log  x,  y ~ a . sin  x,  y ~ a . cos  x y = a*,  are  equations  to 
transcendental  curves  ; and  the  latter  in  particular  is  an  equa- 
tion to  an  exponential  curve. 

Def  4.  Curves  that  turn  round  a fixed  point  or  centre, 
gradually  receding  from  it,  are  called  spiral  or  radial  curves. 

Def.  5.  Family  or  tribe  of  curves,  is  an  assemblage  of 
several  curves  of  different  kinds,  all  defined  by  the  same 
equation  of  an  indeterminate  degree  ; but  differently,  accord- 
ing to  the  diversity  of  their  kind.  For  example,  suppose  an 
equation  of  an  indeterminate  degree,  am~ 1 x = ym  : if  m = 2, 
then  will  ax  = y2  ; if  in  — 3,  then  will  a2x  =■  y3  ; ifm  = 4, 
then  is  a2x  = y*.  &c.  : all  which  curves  are  said  to  be  of  the 
same  family  or  tribe. 

Def.  6.  The  axis  of  a figure  is  a right  line  passing  through 
the  centre  of  a curve,  when  it  has  one  : if  it  bisects  the  ordi- 
nates, it  is  called  a diameter. 

Def.  7.  An  asymptote  is  a right  line  which  continually  ap- 
proaches towards  a curve,  but  never  can  touch  it,  unless  the 
curve  could  be  extended  to  an  infinite  distance. 

Def.  8.  An  abscissa  and  an  ordinate,  whether  right  or 
oblique,  are,  when  spoken  of  together,  frequently  termed  co- 
ordinates. 

Art.  1.  The  most  convenient  mode  of  classing  algebraical 
curves,  is  according  to  the  orders  or  dimensions  of  the  equa- 
tions which  express  the  relation  between  the  co-ordinates. 
For  then  the  equation  for  the  same  curve,  remaining  always 
of  the  same  order  so  long  as  each  of  the  assumed  systems  of 
co-ordinates  is  supposed  to  retain  constantly  the  same  inclina- 
tion of  ordinate  to  abscissa,  while  referred  to  different  points 
of  the  curve,  however  the  axis  and  the  origin  of  the  abscissas, 
or  even  the  inclination  of  the  co-ordinates  in  different  systems, 
may  vary  ; the  same  curve  will  never  be  ranked  under  dif- 
ferent orders,  according  to  this  method.  If  therefore  we 
take,  for  a distinctive  character,  the  number  of  dimensions 
which  the  co-ordinates,  whether  rectangular  or  oblique,  form 
in  the  equation,  we  shall  not  disturb  the  order  of  the  classes, 
by  changing  the  axis  and  the  origin  of  the  abscissas,  or  by  va- 
rying the  inclination  of  the  co-ordinates. 

2 As  algebraists  call  orders  of  different  kinds  of  equations.' 
those  which  constitute  the  greater  or  less  number  of  dimen- 
sions, they  distinguish  by  the  same  name  the  different  kinds 
of  resulting  lines.  Consequently  the  general  equation  of  the 
first  order  being  0 = a.  -J-  fix  -f-  yy  ; we  may  refer  to  the 
first  order  all  the  lines  which,  by  taking  x and  y for  the  co- 
ordinates, whether  rectangular  or  oblique,  give  rise  to  this 

equation. 


NATURE  AND  PROPERTIES  OF  CURVES.  27  b 


equation.  But  this  equation  comprises  the  right  line  alone, 
which  is  the  most  simple  of  all  lines  ; and  since,  for  this  rea- 
son, the  name  of  curve  does  not  properly  apply  to  the  first  or- 
der, we  do  not  usually  distinguish  the  different  orders  by 
the  name  of  curve  lines,  but  simply  by  thq  generic  term  of 
lines  : hence  the  first  order  of  lines  does  not  comprehend  any 
curves,  but  solely  the  right  line. 

As  far  the  rest,  it  is  indifferent  whether  the  co-ordinates 
are  perpendicular  or  not ; for  if  the  ordinates  make  with  the 
axis  an  angle  <p  whose  sine  is  ^ and  cbsine  v,  we  can  refer  the 
equation  to  that  of  the  rectangular  co-ordinates,  by  making 

y — --,  and  * = — 4-  t;  which  will  give  for  an  equation 

1 fJt  (A  ° 

between  the  perpendiculars  t and  u, 

0 = et  -j-  j2t  -f-  ( — -p— ) u. 

ft  (A 

Thus  it  follows  evidently;  that  the  signification  of  the 
[ equation  is  not  limited  by  supposing  the  ordinates  to  be  rightly 
applied  : and  it  will  be  the  same  with  equations  of  superior 
orders,  which  will  not  be  less  general  though  the  co-ordinates 
■ are  perpendicular.  Hence,  since  the  determination  of  the  in- 
clination of  the  ordinates  applied  to  the  axis,  takes  nothing 
from  the  generality  of  a general  equation  of  any  order  what- 
ever, we  put  no  restriction  on  its  signification  by  supposing 
the  co-ordinates  rectangular  ; and  the  equation  will  be  of  the 
same  order  whether  the  co-ordinates  be  rectangular  or  oblique, 
i 3.  All  the  lines  of  the  second  order  will  .be  comprised  in 
I the  general  equation. 

0 = (t  /3  r -}-  yy  -f-  a'x2  -j-e.ry  -f-  f y 2 
that  is  to  say,  we  may  class  among  lines  of  the  second  order 
all  the  curve  lines  which  this  equation  expresses,  x and  y de- 
noting the  rectangular  co-ordinates  These  curve  lines  are 
[therefore  the  most  simple  of  all,  since  there  are  no  curves  in 
the  first  order  of  lines  ; it  is  for  this  reason  that  some  writers 
call  them  curves  of  the  first  order.  But  the  curves  included 
in  this  equation  are.  better  known  under  the  name  of  conic 
sections,  because  they  all  result  from  sections  of  the  cone. 
The  different  kinds  of  these  lines  are  the  ellipse,  the  circle,  , 
or  ellipse  with  equal  axes  , the  parabola,  and  the  hyperbola  ; 
the  properties  of  all  which  may  be  deduced  with  facility  from 
the  preceding  general  equatiou.  Or  this  equation  may  be 
transformed  into  the  subjoined  one  : 


V2  + 


iX+y 


V + 


Sx~  + £x  -f-  a. 


0 


276  NATURE  AND  PROPERTIES  OF  CURVES. 


and  this  again  may  be  reduced  to  the  still  more  simple  fora 
y 2 = fx2  + gx  + h. 

Here,  when  the  first  term  fx2  is  affirmative , the  curve  ex- 
pressed by  the  equation  is  a hyperbola  ; when/r2  is  negative 
the  curve  is  an  ellipse  ; when  that  term  is  absent,  the  curve 
is  a parabola.  When  x is  taken  upon  a diameter,  the  equa- 
tions reduce  to  those  already  given  in  sec.  4 ch.  i. 

The  mode  of  effecting  these  transformations  is  omitted  foi 
the  sake  of  brevity.  This  section  contains  a summary,  not  an 
investigation  of  properties  : the  latter  would  require  many 
volumes,  instead  of  a section. 

4.  Under  lines  of  the  third  order,  or  curves  of  the  second, 
are  classed  all  those  which  may  be  expressed  by  the  equation 
0 = «■  -\-fix-\-yy  -f  $x2  + £xy-\-(^y2  + vx2ffi6x2y-\ -ixy2  -f-  r.y2 . 
And  in  like  manner  we  regard  as  lines  . of  the  fourth  order, 
those  curves  which  are  furnished  by  the  general  equation 
0 = a.  + fix  + yy  + S'x2  + exy  + £y2  + s».t3  + 6x2y  + txy2  -f- 
xy2  -f-  4"  A41 3y  + fx2y2  -f*  ixy2  4~  «y4  ; 

taking  always  x and  y for  rectangular  co-ordinates.  In  the 
most  general  equation  of  the  third  order,  there  are  10  con- 
stant quantities,  and  in  that  of  the  fourth  order  15,  which 
may  be  determined  at  pleasure  ; whence  it  results  that  the 
kinds  of  lines  of  the  third  order,  and,  much  more,  those  of  the 
fourth  order,  are  considerably  more  numerous  than  those 
of  the  second. 


5.  It  will  now  be  easy  to  conceive,  from  what  has  gone  be- 
fore, what  are  the  curve  lines  that  appertain  to  the  fifth,  sixth 
seventh,  or  any  higher  order  ; but  as  it  is  necessary  to  add  to 
the  general  equation  of  the  fourth  order,  the  terms 


Xs,  x*y,  x2y2,  x2 y2 , xy* , ys  , 

with  their  respective  constant  co-efficients,  to  have  the  general 
equation  comprising  all  the  lines  of  the  fifth  order,  this  latter 
will  be  composed  of  21  terms  : and  the  general  equation  com- 
prehending all  the  lines  of  the  sixth  order,  will  have  28  terms  ; 
and  so  on,  conformably  to  the  law  of  the  triangular' numbers. 
Thus  the  most  general  equation  for  lines  of  the  order  >»,  will 

contain  ilf  4-  0 • ( 4-_r_i  terms,  and  as  manv  constant  letters. 


which  may  be  determined  at  pleasure. 


6.  Since  the  order  of  the  proposed  equation  between  the 
co-ordinates,  makes  known  that  of  the  curve  line  ; whenever 
we  have  given  an  algebraic  equation  between  the  co-ordinates 
x and  y,  or  t and  u,  wre  know  at  once  to  what  order  it  is  ne- 
cessary to  refer  the  curve  represented  by  that  equation.  If 
the  equation  be  irrational,  it  must  be  freed  from  radicals,  and 

if 


NATURE  AND  PROPERTIES  OF  CURVES. 


277 


if  there  be  fractions,  they  must  be  made  to  disappear  ; this 
done,  the  greatest  number  of  dimensions  formed  b)  the  va- 
riable quantities  x and  y,  will  indicate  the  order  to  which  the 
line  belongs.  Thus  the  curve  which  is  denoted  by  this  equa- 
tion y2  — ax  = 0,  will  be  of  the  second  order  of  lines,  or  of 
the  first  order  of  curves  ; while  the  curve  represented  by  the 
equation^3  — x^/  (a2  — x2),  will  be  of  the  third  order  (that 
is,  the  fourth  order  of  lines),  because  the  equation  is  of  the 
fourth  order  when  freed  from  radicals  ; and  the  line  which  is 

indicated  by  the  equation  y = ° 3—  will  be  of  the  third 

a 2 T x2 

order,  or  of  the  second  order  of  curves,  because  the  equation 
when  the  fraction  is  made  to  disappear,  becomes  a2y  - f-  x2y  — 
a3  — ax2 , where  the  term  x2y  contains  three  dimensions. 

7.  It  is  possible  that  one  and  the  same  equation  may  give 
different  curves,  according  as  the  applicates  or  ordinates  fall 
upon  the  axis  perpendicularly  or  under  a given  obliquity. 
For  instance,  this  equation,  yz  —ax — x Q,  gives  a circle,  when 
the  co-ordinates  are  supposed  perpendicular  ; but  when  the 
co-ordinates  are  oblique,  the  curve  represented  b}^  the  same 
equation  will  be  an  ellipse.  Yet  all  these  different  curves  ap- 
pertain to  the  same  order,  because  the  transformation  of  rect- 
angular into  oblique  co-ordinates,  and  the  contrary,  does  not 
affect  the  order  of  the  curve,  or  of  its  equation.  Hence, 
though  the  magnitude  of  the  angles  which  the  ordinates  form 
with  the  axis,  neither  augments  nor  diminishes  the  generality 
of  the  equation,  which  expresses  the  lines  of  each  order  ; yet, 
a particular  equation  being  given,  the  curve  which  it  expresses 
can  only  be  determined  when  the  angle  between  the  co-ordi- 
nates is  determined  also. 

8.  That  a curve  line  may  relate  properly  to  the  order  in- 
dicated by  the  equation,  it  is  requisite  that  this  equation  be 
not  decomposable  into  rational  factors  ; for  if  it  could  be  com- 
posed of  two  or  of  more  such  factors,  it  would  then  compre- 
hend as  many  equations,  each  of  which  would  generate  a 
particular  line,  and  the  re-union  of  these  lines  would  be  all 
that  the  equation  proposed  could  represent.  Those  equations, 
then,  which  may  be  decomposed  into  such  factors,  do  not 
comprise  one  continued  curve,  but  several  at  once,  each  of 
which  may  be  expressed  by  a particular  equation  ; and  such 
combinations  of  separate  curves  are  denoted  by  the  term  com- 
plex curves. 

Thus,  the  equation  y 1 = ay  4-  xy  — ax,  which  seems  to 
appertain  to  a line  of  the  second  order,  if  it  be  reduced  to 
zero  by  making  y 8 ■—  ay  — xy  -j-  ax  — 0,  will  be  composed 
of  the  factors  (y  — x)  (y  — a)  = 0 ; it  therefore  comprises 

the 


278  NATURE  AND  PROPERTIES  OF  CURVES. 


the  two  equations  y — x — U,  and  y — a — 0,  both  of  which 
belong  to  the  light  line  : the  first  forms  with  the  'axis  at  the 
origin  of  the  abcissas  an  angle  equal  to  half  a rignt  angle  ; 
and  the  second  is  parallel  to  the  axis,  and  drawn  at  a distance 
= a.  These  two  lines,  considered  together,  are  comprized 
in  the  proposed  equation  y2  — ay  + xy — ax.  In  like  man- 
ner we  may  regard  as  complex  this  equation  y4  — xy3  — 
a2  x2  — ay3-\-ax2y  -\-a2  xy  = 0 ; for  its  factors  being  (y—x) 

( y — «)  ( y 2 — ax)  — 0,  instead  of  denoting  one  continued  line 
of  the  fourth  order,  it  comprizes  three  distinct  lines,  viz.  two 
right  lines,  and  one  curve  denoted  by  the  equa.  y2  — ax  — 0. 

9.  We  may  therefore  form  at  pleasure  any  complex  lines 
whatever,  which  shall  contain  2 or  more  right  lines  or  curves 
For,  if  the  nature  of  each  line  is  expressed  by  an  equation  re- 
ferred to  the  same  axis,  and  to  the  same 
origin  of  the  abscissas,  and  after  having 
reduced  each  equation  to  zero,  we  mul- 
tiply them  one  by  another,  there  will 
result  a complex  equation  which  at  once 
comprizes  all  the  lines  assumed.  For 
example,  if  from  the  centre  c,  with  a 
radius  ca  = a,  a circle  be  described  ; and  further,  if  a right 
line  ln  be  drawn  through  the  centre  c ; then  we  may,  for  any 
assumed  axis,  find  an  equation  which  will  at  once  include  the 
circle  and  the  right  line,  as  though  these  two  lines  formed  on- 
ly one. 

Suppose  there  be  taken  for  an  axis  the  diameter  ab,  that 
forms  with  the  right  line  ln  an  angle  equal  to  half  a right 
angle  : having  placed  the  origin  of  the  abscissas  in  a,  make 
the  abscissa  ab  = x,  and  the  applicate  or  ordinate  pm  = y ; 
we  shall  have  for  the  right  line,  pm  — cp  = a — x ; and  since 
the  point  m of  the  right  line  falls  on  the  side  of  those  ordi- 
nates which  arc  reckoned  negative,  we  have  y = — a + x, 
or?/ — x + a ~ 0 : but,  for  the  circle,  we  have  pm2  = ap  . pb, 
and  bp  — 2a  — x,  which  gives  y 2 — 2 ax  — x2 , or  y2  -f-  x2  — 

2 ax  — 0.  Multiplying  these  two  equations  together  we  obtain  * 
the  complex  equation  of  the  third  order, 

^3  __  y-x  -f-  yx2  — x3  ~r  ay2  — 2 axy  + 3ax2  — 2a2 x — 0, 

which  represents,  at  once,  the  circle  and  the  right  line  Hence, 
we  shall  tind  that  to  the  abscissa  ap  = x,  corresponds  three 
ordinates,  namely,  two  for  the  circle,  and  one  for  the  right 
line.  Let,  for  example,  x — |a,  the  equation  will  become 
y3^iay2  - ia2y — fa3  = 0 ; whence  we  first  findy  + 
and  by  dividing  by  this  root  we  obtain  y2  — |a2  = 0,  the  two 
roots  of  which  being  taken  and  ranked  with  the  former,  give 
the  three  following  values  of  y : 


V 


IB ^ 


I.  y = 


NATURE  AND  PROPERTIES  OF  CURVES.  279 


I-  y = — \a. 

II.  y p -j-  \a^/ 3. 

ID.  y — — i-a^/3. 

tVe  see  therefore  that  the  whole  is  represented  by  one  equa- 
tion, as  if  the  circle  together  with  the  right  line  formed  only 
one  continued  curve. 


10.  This  difference  between  simple  and  complex  curves 
being  once  established,  it  is  manifest  that  the  lines  of  the  se- 
cond order  are  either  continued  curves,  or  complex  lines  form- 
ed of  two  right  lines  ; for  if  the  genera]  equation  have  ra- 
tional factors,  they  must  be  of  the  first  order,  and  consequent- 
ly will  denote  right  lines.  Lines  of  the  third  order  will  be 
either  simple,  or  complex,  formed  either  of  a right  line  and  a 
line  of  the  second  order,  or  of  three  right  lines.  In  like 
manner,  lines  of  the  fourth  order  will  be  continued  and  sim- 
ple, or  complex,  comprizing  a right  line  and  a line  of  the 
third  order,  or  two  lines  of  the  second  order,  or  lastly,  four 
right  lines.  Complex  lines  of  the  fifth  and  superior  orders 
will  be  susceptible  of  an  analogous  combination,  and  of  a 
similar  enumeration.  Hence  it  follows,  that  any  order  what- 
ever of  lines  may  comprize,  at  once,  all  the  lines  of  inferior 
order,  that  is  to  say,  that  they  may  contain  a complex  line  of 
any  inferior  orders  with  one  or  more  right  lines,  or  with  lines 
of  the  second,  third,  &c.  orders  ; so  that  if  we  sum  the  num- 
bers of  each  order,  appertaining  to  the  simple  lines,  there  will 
result  the  number  indicating  the  order  of  the  complex  line. 

Def.  9.  That  is  called  an  hyperbolic  leg,  or  branch  of  a 
curve,  which  approaches  constantly  to  some  asymptote  ; and 
that  a parabolic  one  which  has  no  asymptote. 

Art.  11.  All  the  legs  of  curves  of  the  second  and  higher 
kinds,  as  well  as  of  the  first,  infinitely  drawn  out,  will  be  of 
either  the  hj'perbolic  or  the  parabolic  kind  : and  these  legs 
are  best  known  from  the  tangents.  For  if  the  point  of  con- 
tact be  at  an  infinite  distance,  the  tangent  of  a hyperbolic  leg- 
will  coincide  with  the  asymptote,  and  the  tangent  of  a para- 
bolic leg  will  recede  in  infinitum , will  vanish  and  be  no  where 
found.  Therefore  the  asymptote  of  any  leg  is  found  by  seek- 
ing the  tangent  to  that  leg  at  a point  infinitely  distant:  and 
the  course,  or  way  of  an  infinite  leg,  is  found  by  seeking  the 
position  of  any  right  line  which  is  parallel  to  the  tangent 
where  the  point  of  contact  goes  off  in  infinitum  : for  this  right 
line  is  directed  the  same  way  with  the  infinite  leg. 


Sir  Isaac  Newton's  Reduction  of  all  Lines  of  the  Third  (Dr- 
ier to  four  Cases  of  Equations  ; with'  the  Enumeration  of  those 

CASK  !. 


jh'nrs. 


280  LINES  OF  THE  THIRD  ORDER, 

CASE  I. 

12.  All  the  lines  of  the  first,  third,  fifth,  and  seventh  order,  j 
or  of  any  odd  order,  have  at  least  two  legs  or  sides  proceed-  j 
ing  on  ad  infinitum , and  towards  contrary  parts.  And  all  lines  I 
of  the  third  order  have  to  such  legs  or  branches  running  out  I 
contrary  ways,  and  towards  which  no  other  of  their  infinite 
legs  (except  in  the  Cartesian  parabola)  tend.  If  the  legs  are 
of  the  hyperbolic  kind,  let  gas  be  their  asymptote  ; and  to  it 


let  the  parallel  cbc  be  drawn,  terminated  (if  possible)  at  both  j 
ends  at  the  curve.  Let  this  parallel  be  bisected  in  x,  and 
then  will  the  locus  of  that  point  x be  the  conical  or  common 
hyperbola  xq,  one  of  whose  asymptotes  is  as.  Let  its  other 
asymptote  be  ab.  Then  the  equation  by  which  the  relation 
between  the  ordinate  bc  = y,  and  the  abscissa  ab  = x,  is 
determined,  will  always  be  of  this  form  : viz. 

x y2  -f - ey  = ax3  -f-  bx2  -f-  cx  -J-  d . . (I.) 

Here  the  coefficients  e,  a,  b,  c,  d,  denote  given  quantities, 
affected  with  their  signs  + and  — , of  which  terms  any  one 
may  be  wanting,  provided  the  figure  through  their  defect  does 
not  become  transformed  into  a conic  section.  The  conical 
hyperbola  xq  may  coincide  with  its  asymptotes,  that  is,  the 
point  x may  come  to  be  in  the  line  ab  ; and  then  the  term  + 
ey  will  be  wanting. 

CASE  II. 

13.  But  if  the  right  line  cbc  cannot  be  terminated  both 
ways  at  the  curve,  but  will  come  to  it  only  in  one  point  ; then 
draw  any  line  in  a given  position  which  shall  cut  the  asymp- 
tote as  in  a j as  also  any  other  right  line,  as  bc,  parallel  to 

the 


LINES  OF  THE  THIRD  ORDER. 


231 


the  asymptote,  and  meeting  the  curve  in  the  point  c ; then  the 
equation,  by  which  the  relation  between  the  ordinate  bc  and 
the  abscissa  ab  is  determined,  will  always  assume  this  form  • 
viz.  xy  = ax 3 -j-  bx3  + cx  + d . . . . (II.) 

CASE  III. 

14.  If  the  opposite  legs  be  of  the  parabolic  kind,  draw  the 
right  line  cbc,  terminated  at  both  ends  (if  possible)  at  the 
curve,  and  running  according  to  the  course  of  the  legs  ; which 
line  bisect  in  b : then  shall  the  locus  of  b be  a right  line.  Let 
that  right  line  be  ab,  terminated  at  any  given  point,  as  a : 
then  the  equation,  by  which  the  relation  between  the  ordinate 
bc  and  the  abscissa  ab  is  determined,  will  always  be  of  this 
form  : y 3 = ax3  + bx2  + cx  + d . . . . (III.) 

CASE  IV. 

15.  If  the  right  line  cbc  meet  the  curve  only  in  one  point, 
and  therefore  cannot  be  terminated  at  the  curve  at  both  ends  ; 
let  the  poiut  where  it  comes  to  the  curve  be  c,  and  let  that 
right  line  ai  the  point  b,  fall  on  any  other  right  line  given  in 
position,  as  ab,  and  terminated  at  any  gi.ven  point,  as  a. 
Tnen  w ; the  equation  expressing  the  relation  between  bc 
and  ab,  assume  this  form  : 

y = ax3  + bx2  -f-  cx  + d . . . . (IV.) 

1(S.  In  the  first  case,  or  that  of  equation  i,  if  the  term  ax3 
be  affirmative,  the  figure  will  be  a triple  hyperbola  with  six 
hyperbolic  legs,  which  will  run  on  infinitely  by  the  three 
asymptotes,  of  which  none  are  parallel,  two  legs  towards  each 
asymptote,  and  towards  contrary  parts  ; and  these  asymptotes, 
if  the  term  bx 2 be  not  wanting  in  the  equation,  will  mutually 
intersect  each  other  in  3 points,  forming  thereby  the  triangle 
ndS1.  But  if  the  term  bx2  be  wanting,  they  will  all  converge 
to  the  same  point.  This  kind  of  hyperbola  is  called  redund- 
ant, because  it  exceeds  the  conic  hyperbola  in  the  number  of 
its  hyperbolic  legs. 

In  every  redundant  hyperbola,  if  neither  the  term  ey  be 
wanting,  nor  l2’  — 4 ac  = ae^/a,  the  curve  will  have  no  dia- 
meter ; but  if  either  of  those  occur  separately,  it  will  have 
only  one  diameter  ; and  three , if  they  both  happen.  Such 
diameter  will  always  pass  through  the  intersection  of  two  of 
the  asymptotes,  and  bisect  all  right  lines  which  are  terminated 
each  way  by  those  asymptotes,  and  which  are  parallel  to  the 
third  asymptote. 

17.  If  the  redundant  hyperbola  have  no  diameter,  let  the 
four  roots  or  values  of  x in  the  equation  ax 4 + bx2  -f"  cx2  -f- 
dx  -}-  Ae2  = 0,  be  sought  ; and  suppose  them  to  be  ap,  aw. 
VoL.  II.  37  ATT, 


282 


LINES  OP  THE  THIRD  ORDER. 


A! r,  and  a p (see  the  preceding  figure).  Let  the  ordinates  i*t, 
sett,  7rl,  pt,  be  erected  ; they  shall  touch  the  curve  in  the  points 
T,  r,  7,  t,  and  by  that  contact  shall  give  the  limits  of  the  curve, 
by  which  its  species  will  be  discovered. 

Thus,  if  all  the  roots  ap,  a sr,  asv, Ap,  be  real,  and  have  the 
same  sign,  and  are  unequal,  the  curve  will  consist  of  three 
hyperbolas  and  an  oval  : viz.  an  inscribed  hyperbola  as  ec  ; a 
circumscribed  hyperbola,  as  T 2c;  and  ambigeneal  hyperbola,  (i.  e. 
lying  within  one  asymptote  and  beyond  another)  as  pt ; and 
an  oval  tJ.  This  is  reckoned  the  first  species.  Other  rela- 
tions of  the  roots  of  the  equation,  give  8 more  different  spe- 
cies of  redundant  hyperbolas  without  diameters  ; 12  each  with 
but  one  diameter  ; 2 each  with  three  diameters  ; and  9 each 
with  three  asymptotes  converging  to  a common  point.  Some 
of  these  have  ovals,  some  points  of  decussation,  and  in  some 
the  ovals  degenerate  into  nodes  or  knots. 

18.  When  the  term  ax3  in  equa.  i,  is  negative,  the  figure 
expressed  by  that  equation,  will  be  a deficient  or  defective 
hyperbola  ; that  is,  it  will  have  fewer  legs  than  the  complete 
conic  hyperbola.  Such  is  the  marginal 
figure,  representing  Newton’s  33d  spe- 
cies ; which  is  constituted  of  an  angui- 
neal  or  serpentine  hyperbola,  (both  legs 
approaching  a common  asymptote  by 
means  of  a contrary  flexure,  and  a con- 
jugate oval.  There  are  6 species  of  de- 
fective hyperbolas,  each  having  but  one 
asymptote,  and  only  two  hyperbolic  legs, 
running  out  contrary  ways,  ad  infini- 
tum ; the  asymptote  being  the  first  and  principal  ordinate 
When  the  term  ey  is  not  absent,  the  figure  will  have  no 
diameter  ; when  it  is  absent,  the  figure  will  have  one  diame- 
ter. Of  this  latter  class  there  are  7 different  species,  one  of 
which,  namely  Newton’s  40th  species,  is  exhibited  in  the 
margin. 

19.  If,  in  equation  i,  the  term  ax 3 be 
wanting,  but  bx-  not,  the  figure  ex- 
pressed by  the  equation  remaining,  will 
be  a parabolic  hyperbola,  having  two 
hyperbolic  legs  to  one  asymptote,  and 
two  parabolic  legs  converging  one  and 
the  same  way.  When  the  term  ey  is 
not  wanting,  the  figure  will  have  no 
diameter  \ if  that  term  be  wanting,  the 
figure  will  have  one  diameter.  There  are  7 species  apper- 
taining to  the  former  case  ; and  4 to  the  latter. 

20.  When 


LINES  OF  THE  THIRD  ORDER. 


283 


20.  When,  in  equa.  i,  the  terms  cu3,  bxa~,  are  wanting,  or 
when  that  equation  becomes  xy2  + ey  = cx  -}-  d,  it  expresses 
a figure  consisting  of  three  hyperbolas  opposite  to  one  an- 
other, one  lying  between  the  parallel  asymptotes,  and  the 
other  two  without  : each  of  these  curves  having  three  asymp- 
totes, one  of  which  is  the  first  and 
principal  ordinate,  the  other  two  pa- 
rallel to  the  abscissa,  and  equally 
distant  from  it  : as  in  the  annexed 
figure  of  Newton’s  60th  species. 

Otherwise  the  said  equation  ex- 
presses two  opposite  circumscribed 
hyperbolas,  and  an  anguineal  hyper- 
bola between  the  asymptotes-  Under 
this  class  there  are  4 species,  called 
by  Newton  Hyperbolisnue  of  an  hyperbola.  By  hyperbolismae 
of  a figure  he  means  to  signify  when  the  ordinate  comes  out, 
by  dividing  the  rectangle  under  the  ordinate  of  a given  conic 
section  and  a given  right  line,  by  the  common  abscissa. 

21-  When  the  term  c r2  is  negative,  the  figure  expressed 
by  -the  equation  xy2  + ey  = — cx 2 -f-  d , is  either  a serpentine 
hyperbola,  having  only  one  asymptote,  being  the  principal 
ordinate  ; or  else  it  is  a conchoidal  figure.  Under  this  class 
there  are  3 species,  called  Hyperbolisrn.ee  of  an  ellipse. 

22.  When  the  term  c.r2  is  absent,  the  equa.  xy2  -f-  ey  — d, 
expresses  two  hyperbolas,  lying,  not  in  the  opposite  angles  of 
the  asymptotes  (as  in  the  conic  hyperbola),  but  in  the  adja- 
cent angles.  Here  there  are  only  2 species,  one  consisting  of 
an  inscribed  and  an  ambigeneal  hyperbola,  the  other  of  two 
inscribed  hyperbolas  These  two  species  are  called  the  Hy- 
perbolismce  of  a parabola. 

23.  In  the  second  case  of  equations,  or  that  of  equation  ir, 
there  is  but  one  figure  ; which  has  four  infinite  legs.  Of 
these,  two  are  hyperbolic  about  one  asymptote,  tending  to- 
wards contrary  parts,  and  two  converging  parabolic  legs, 
making  with  the  former  nearly  the  figure  of  a trident , the 
familiar  name  given  to  this  species.  This  is  the  Cartesian 
parabola,  by  which  equations  of  6 dimensions  are  sometimes 
constructed  : it  is  the  66th  species  of  Newton’s  enumeration. 

24.  The  third  case  of  equations,  or 
equa  hi,  expresses  a figure  having  two 
parabolic  legs  running  out  contrary  ways: 
of  these  there  are  5 different  species, 
called  diverging  or  bell-form  parabolas  ; 
of  which  2 have  ovals,  1 is  nodate,  1 

, punctate,  and  1 cuspidate.  The  figure  shows  Newton’s  67th 

species  ; 


284 


LINES  OF  THE  FOURTH  ORDER. 


species  ; in  which  the  oval  must  always  be  so  small  that  n« 
right  line  which  cuts  it  twice  can  cut  the  parabolic  curve  c£ 
more  than  once. 

25.  In  the  case  to  which  equa.  iv 
refers,  there  is  but  one  species.  It  ex- 
presses the  cubical  parabola  with  con- 
trary legs.  This  curve  may  easily  be 
described  mechanically  by  means  of  a 
square  and  an  equilateral  hyperbola.  Its 
most  simple  property  is,  that  rm  (paral- 
lel to  aq)  always  varies  a6  qn3 — qr3. 

26.  Thus  according  to  Newton  there  are  72  species  of  lines 
of  the  third  order.  But  Mr.  Stirling  discovered  four  more 
species  of  redundant  hyperbolas  ; and  Mr.  Stone  two  more 
species  of  deficient  hyperbolas,  expressed  by  the  equation 
yx3  = bxs  + cx  -f-  d : i.  e.  in  the  case  when  bx2  -\-cx-\-d=0, 
has  two  unequal  negative  roots,  and  in  that  where  the  equa- 
tion has  two  equal  negative  roots.  So  that  there  are  at  least 
78  different  species  of  lines  of  the  third  order.  Indeed  Euler, 
who  classes  all  the  varieties  of  lines  of  the  third  order  under 
16  general  species  affirms  that  they  comprehend  more  than 
80  varieties  ; of  which  the  preceding  enumeration  necessarily 
comprizes  nearly  the  whole. 

27.  Lines  of  the  fourth  order  are  divided  by  Euler  into 
146  classes  ; and  these  comprize  more  than  5000  varieties  : 
they  all  flow  from  the  different  relations  of  the  quantities  in 
the  10  general  equations  subjoined. 

1-yl  +fx2y2  f g*ys  + /i*2y+iy 2 -f  h*y+ly-\ 

%-y*  +fxy2  +gX2y-\~,>xy2  -f  ixy  + ky  . . | 

3.  *2ys+fy3  -i-gxsy-f-  by2  +ky  ■ . 

4.  ,v»y2  +_/j,3  +gy2  +hxy  +iy  . . 

5. y2  -f ifxyv  +gX2y  + hy  . . 

6.  y3  +/'xy2  +gxy  -\-hy  ■ . • • 

7. yi  4-  ex3y+fxy2-[-gXy2  -f  Ay2+ixy-f  ky  \ 

Z.xzy  + exyi+fxsy+gy2  -f  hxy  + iy  . . / axs+bx2  +>x+  J 
9.  x*y+ey*  +fxy*+gxy  + hy  . . «/. 

l0.x3y-{-  ey3  -\-fy2  "\-gxy  +ky  '•  • • -* 

28.  Lines  of  the  fifth  and  higher  orders,  of  necessity  be- 
come still  more  numerous  ; and  present  too  many  varieties  to 
admit  of  any  classification,  at  least  in  moderate  compass. 
Instead,  therefore,  of  dwelling  upon  these  ; we  shall  give  a 
concise  sketch  of  the  most  curious  and  important  properties 
of  curve  lines  in  general,  as  they  have  been  deduced  from  a 
contemplation  of  tbe  nature  and  mutual  relation  of  the  roots 
of  the  equations  representing  those  curves.  Thus  a curve 
being  called  of  n dimensions,  or  a line  of  the  nth  order  when 
its  representative  equation  rises  to  n dimensions  ; then  since 

for 


)>=ax  4 tfixs  *exz  sdxta. 


GENERAL  PROPERTIES  OF  CURVES. 


285 


for  every  different  value  of  x there  are  n values  of  y,  it  will 
commonly  happen  that  the  ordinate  will  cut  the  curve  in  n or 
in  n — 2,  n — 4,  &c.  points,  according  as  the  equation  has 
»,  orrt  — 2,  n — 4,  &c  possible  roots.  It  is  not  however  to 
be  inferred  that  a right  line  will  cut  a curve  of  n dimensions, 
in  n,  n — 2,  n — 4,  &c.  points  only ; for  if  this  were  the 
case,  a line  of  the  2d  order,  a conic  section  for  instance,  could 
only  be  cut  by  a right  line  in  two  points  ; — but  this  is  raani 
festly  incorrect,  for  thbugh  a conic  parabola  will  be  cut  in  two 
points  by  a right  line  oblique  to  the  axis,  yet  a right  line  pa- 
rallel to  the  axis  can  only  cut  the  curve  in  one  point. 

29.  It  is  true  in  general,  that  lines  of  the  n order  cannot 
be  cut  by  a right  line  in  more  than  n points  ; but  it  does  not 
hence  follow,  that  any  right  line  whatever  will  cut  in  n points 
every  line  of  that  order  ; it  may  happen  that  the  number  of 
intersections  is  n — 1,  n — 2,  n — 3,  &c.  to  n — n.  The 
number  of  intersections  that  any  right  line  whatever  makes, 
with  a given  curve  line  cannot  therefore  determine  the  order 
to  which  a curve  line  appertains.  For,  as  Euler  remarks,  if 
the  number  of  intersections  be  = n,  it  does  not  follow  that 
the  curve  belongs  to  the  n order,  but  it  may  be  referred  to 
some  superior  order ; indeed  it  may  happen  that  the  curve  is 
not  algebraic,  but  transcendental.  This  case  excepted,  how- 
ever, Euler  contends  that  we  may  always  affirm  positively 
that  a curve  line  which  is  cut  by  a right  line  in  n points,  can- 
not belong  to  an  order  of  lines  inferior  to  n.  Thus,  when  a 
right  line  cuts  a curve  in  4 points,  it  is  certain  that  the  curve 
does  not  belong  to  either  the  second  or  third  order  of  lines  ; 
but  whether  it  be  referred  to  the  fourth,  or  a superior  order, 
or  whether  it  be  transcendental,  is  not  to  be  decided  but  by 
analysis. 

30.  Dr.  Waring  has  carried  this  enquiry  a step  further  than 
Euler,  and  has  demonstrated  that  there  are  curves  of  any  num- 
ber of  odd  orders,  that  cut  a right  line  in  2,  4,  G,  &c.  points 
only  ; and  of  any  number  of  even  orders  that  cut  a right  line 
in  3,  5,  7,  &c.  points  only  ; whence  this  author  likewise  in- 
fers, that  the  order  of  the  curve  cannot  be  announced  from 
the  number  of  points  in  which  it  cuts  a right  line.  See  his 
Proprietates  Algebraicarum  Curvarum. 

31.  Every  geometrical  curve  being  continued,  either  re- 
, turns  into  itself,  or  goes  on  to  an  infinite  distance.  And  if  any 

plane  curve  has  two  infinite  branches  or  legs,  they  join  one 
another  either  at  a finite,  or  at  an  infinite  distance. 

32.  In  any  curve,  if  tangents  be  drawn  to  all  points  of  the 
curve  ; and  if  they  always  cut  the  abscissa  at  a finite  distance 
from  its  origin  ; that  curve  has  an  asymptote,  otherwise,  not, 

33.  A 


286 


GENERAL  PROPERTIES  OF  CURVES. 


33.  A line  of  any  order  may  have  as  many  asymptotes  as  it 
has  dimensions,  and  no  more. 

34.  An  asymptote  may  intersect  the  curve  in  so  many 
points  abating  two,  as  the  equation  of  the  curve  has  dimen- 
sions. Thus,  in  a conic  section,  which  is  the  second  order 
of  lines,  the  asymptote  does  not  cut  the  curve  at  all  ; in  the 
third  order  it  can  only  cut  it  in  one  point ; in  the  fourth  order 
in  two  points  ; and  so  on. 

35.  If  a curve  have  as  many  asymptotes,  as  it  has  dimensions, 
and  a right  line  be  drawn  to  cut  them  all,  the  parts  of  that 
measured  from  the  a«3unptotes  to  the  curve,  will  together  be 
equal  to  the  parts  measured  in  the  same  direction,  from  the 
curve  to  tha  asymptotes 

36.  If  a curve  of  n dimensions  have  n asymptotes,  then  the 
content  of  the  n abscissas  will  be  to  the  content  of  the  n or- 
dinates, in  the  same  ratio  in  the  curve  and  asymptotes  ; the 
sum  of  thejr  n subnormals,  to  ordinates  perpendicular  to  their 
abscissas,  will  be  equal  to  the  curve  and  the  asymptotes  ; and 
they  will  have  the  same  central  and  diametral  curves. 

37.  If  two  curves  of  n and  m dimensions  have  a common 
asymptote  ; or  the  terms  of  the  equations  to  the  curves  of  the 
greatest  dimensions  have  a common  divisor  ; then  the  curves 
cannot  intersect  each  other  in  n X in  points,  possible  or  im- 
possible. If  the  two  curves  have  a common  general  centre, 
and  intersect  each  other  in  n X m points,  then  the  sum  of 
the  affirmative  abscissas,  &c.  to  those  points,  will  be  equal  to 
the  sum  of  the  negative  ; and  the  sum  of  the  n subnormals 
to  a curve  which  has  a general  centre,  will  be  proportional 
to  the  distance  from  that  centre. 

38.  Lines  of  the  third,  fifth,  seventh,  &c.  order,  or  any 
odd  number,  have,  as  before  remarked,  at  least  two  infinite 
legs  or  branches,  running  contrary  ways  ; while  in  lines  of  the 
second,  fourth,  sixth,  or  any  even  number  of  dimensions,  the 
figure  may  return  into  itself,  and  be  contained  within  certain 
limits, 

39.  If  the  right  lines  ap,  pm,  forming  a given  angle,  af.m 
cut  a geometrical  line  of  any  order  in  as  many  points  as  it 
has  dimensions,  the  product  of  the  segments  of  the  first  ter- 
minated by  p and  the  curve,  will  always  be  to  the  product  of 
the  segments  of  the  latter,  terminated  by  the  same  point  and 
the  curve,  in  an  invariable  ratio. 

40.  With  respect  to  double,  triple,  quadruple,  and  other 
multiple  points,  or  the  points  of  intersection  of  2,  3,  4,  or 
more  branches  of  a curve,  their  nature  and  number  may  be 
estimated  by  means  of  the  following  principles.  1.  A curve  of 
the  n order  is  determinate  when  it  is  subjected  to  pass  through 


GENERAL  PROPERTIES  OF  CURVES. 


287 


the  number 


-r  (n+2) 


— 1 points. 


2.  A curve  of  the  n 


order  cannot  intersect  a curve  of  the  m order  in  more  than 
mn  points. 

Hence  it  follows  that  a curve  of  the  second  order,  for  ex- 
ample, can  always  pass  through  5 given  points  (not  in  the 
same  right  line),  and  cannot  meet  a curve  of  them  order  in 
more  than  mn  points  ; and  it  is  impossible  that  a curve  of  the 
m order  should  have  5 points  whose  degrees  of  multiplicity 
make  together  more  than  2m  points.  Thus  a line  of  the 
fourth  order  cannot  have  four  double  points  ; because  the  line 
of  the  second  order  which  would  pass  through  these  four 
double  points,  and  through  a fifth  simple  point  of  the  curve 
of  the  fourth  dimension,  would  meet  9 times  ; which  is  im- 
possible, since  there  can  only  be  an  intersection  2 X 4 or  8 
times. 

For  the  same  reason,  a curve  line  of  the  fifth  cannot,  with 
one  triple  point,  have  more  than  three  double  points  : and  in 
a similar  manner  we  may  reason  for  curves  of  higher  orders. 

Again,  for  the  known  proposition,  that  we  can  always 
make  a line  of  the  third  order  pass  through  nine  points,  and 
that  a curve  of  that  order  cannot  meet  a curve  of  the  m order 
in  more  that  3m  points,  we  may  conclude  that  a curve  of  the 
m order  cannot  have  nine  points,  the  degrees  of  multiplicity 
of  which  make  together  a number  greater  than  3m.  Thus, 
a line  of  the  fifth  order  cannot  have  more  than  6 double 
points  ; a line  of  the  6th  order,  which  cannot  have  more  than 
one  quadruple  point,  cannot  have  with  that  quadruple  point 
more  than  6 double  points  ; nor  with  two  triple  points  more 
than  3 double  points  ; nor  even  with  one  triple  point  more 
than  7 double  points.  Analogous  conclusions  obtain  with 
respect  to  a line  of  the  fourth  order,  which  we  may  cause  to 
pass  through  14  points,  and  which  can  only  meet  a curve  of 
the  m order  in  4m  points,  and  so  on. 


41.  The  properties  of  curves  of  a superior  order,  agree, 
under  certain  modifications,  with  those  of  all  inferior  orders. 
For  though  some  line  or  lines  become  evanescent,  and  others 
become  infinite,  some  coincide,  others  become  equal  ; some 
points  coincide,  and  others  are  removed  to  an  infinite  dis- 
tance ; yet,  under  these  circumstances  the  general  properties 
still  hold  good  with  regard  to  the  remaining  quantities  ; so 
that  whatever  is  demonstrated  generally  of  any  order,  holds 
true  in  the  inferior  orders  : and,  on  the  contrary,  there  is 
hardly  any  property  of  the  inferior  orders,  but  there  is  some 
similar  to  it,  in  the  superior  ones. 

For,  as  in  the  conic  sections,  if  two  parallel  lines  are  drawn 

to 


283 


GENERAL  PROPERTIES  OF  CURVES. 


to  terminate  at  the  section,  the  right  line  that  bisects  these 
will  bisect  all  other  lines  parallel  to  them  ;*  and  is  therefore 
called  a diameter  of  the  figure,  and  the  bisected  lines  ordi- 
natesand  the  intersections  of  the  diameter  with  the  curve 
v erticis  ; the  common  intersection  of  all  the  diameters  the 
centre  ; and  that  diameter  which  is  perpendicular  to  the  or- 
dinates, the  vertex.  So  likewise  in  higher  curves,  if  two 
parallel  lines  be  drawn,  each  to  cut  the  curve  in  the  number 
of  points  that  indicate  the  order  of  the  curve  ; the  right  line 
that  cuts  these  parallels  so,  that  the  sum  of  the  parts  on  one 
side  of  the  line,  estimated  to  the  curve,  is  equal  to  tbe  sum 
of  the  parts  on  the  other  side,  it  will  cut  in  tbe  same  man-- 
ner  all  other  lines  parallel  to  them  that  meet  the  curve  in  the 
same  number  of  points  ; in  this  case  also  the  divided  lines  are 
called  ordinates,  the  line  so  dividing  them  a diameter,  the 
intersection  of  the  diameter  and  the  curve  vertices  ; the  com- 
mon intersection  of  two  or  more  diameters  the  centre  ; the 
diameter  perpendicular  to  the  ordinates,  if  there  be  any  such, 
the  axis  ; and  when  all  the  diameters  concur  in  one  point,  that 
is  the  general  centre. 

Again,  the  conic  hyperbola,  being  a line  of  the  second 
order,  has  two  asymptotes  ; so  likewise,  that  of  the  third 
order  may  have  three  ; that  of  the  fourth,  four  ; and  so  on  ; 
and  they  can  have  no  more  And  as  the  parts  of  any  right  line 
between  the  hyperbola  and  its  asymptotes  are  equal  ; so  like- 
wise in  the  third  order  of  lines,  if  any  line  be  drawn  cutting 
the  curve  and  its  asymptotes  in  three  points  ; the  «um  of  two 
parts  of  it  falling  the  same  way  from  the  asymptotes  to  the 
curve,  will  be  equal  to  the  part  falling  the  contrary  way  from 
the  third  asymptote  to  the  curve  ; and  so  of  higher  curves. 

Also,  in  the  conic  sections  which  are  not  parabolic  : as  the 
square  of  the  ordinate,  or  the  rectangle  of  the  parts  of  it  on 
each  side  of  the  diameter,  is  to  the  rectangle  of  the  parts  of 
the  diameter,  terminating  at  the  vertices,  in  a constant  ratio, 
viz.  that  of  the  latus  rectum,  to  the  transverse  diameter.  So 
in  non-parabolic  curves  of  the  next  superior  order,  the  solid 
under  the  three  ordinates,  is  to  the  solid  under  the  three  ab- 
scissas, or  the  distances  to  the  three  vertices ; in  a certain  given 
ratio.  In  which  ratio  if  there  be  taken  three  lines  propor- 
tional to  the  three  diameters,  each  to  each  ; then  each  of  these 
three  lines  may  be  called  a latus  rectum , and  each  of  the  cor- 
responding diameters  a transverse  diameter.  And,  in  the 
common,  or  Apollonian  parabola,  which  has  but  one  vertex 
for  one  diameter  the  rectangle  of  the  ordinates  is  equal  to 
the  rectangle  of  the  abscissa  and  latus  rectum  ; so,  in  those 
curves  of  the  second  kind,  or  lines  of  the  third  kind  which 

have 


NATURE  AND  EQUATIONS  OF  CURVES.  289 


have  only  two  vertices  to  the  same  diameter,  the  solid  under 
the  three  ordinates,  is  equal  to  the  solid  under  the  two  ab- 
scissas, and  a given  line,  which  may  be  reckoned  the  latus 
rectum. 

Lastly,  since  in  the  conic  sections  where  two  parallel  lines 
terminating  at  the  curve  both  ways,  are  cut  by  two  other  pa- 
rallels likewise  terminated  by  the  curve  ; we  have  the  rect- 
angle of  the  parts  of  one  of  the  first,  to  the  rectangle  of  the 
parts  of  one  of  the  second  lines,  as  the  rectangle  of  the  parts 
of  the  second  of  the  former,  to  the  rectangle  of  the  parts  of 
the  second  of  the  latter  pair  passing  also  through  the  com- 
mon point  of  their  division.  So.  when  four  such  lines  are 
drawn  in  a curve  of  the  second  kind,  and  each  meeting  it  in 
three  points  ; the  solid  under  the  parts  of  the  first  line,  will 
be  to  that  under  the  parts  of  the  third,  as  the  solid  under  the 
parts  of  the  second,  to  that  under  the  parts  of  the  fourth. 
And  the  analogy  between  curves  of  different  orders  may  be 
carried  much  further  : but  as  eaough  is  givea  for  the  objects 
of  this  work  ; we  shall  now  present  a few  of  the  most  useful 
problems. 

PROBLEM  I. 

Knowing  the  Characteristic  Property,  or  the  Manner  of  De 
scription  of  a Curve,  to  find  its  Equation. 

! This  in  most  cases  will  be  a matter  of  great  simplicity  ; be- 
cause  the  manner  of  description  suggests  ihe  relation  between 
the  ordinates  and  their  corresponding  abscissas  ; and  this  re- 
lation when  expressed  algebraically,  is  no  other  than  the  equa- 
tion to  the  curve.  Examples  of  this  problem  have  already  oc- 
curred in  sec.  4 of  vol.  1 -.  to  which  the  following  are  now 
added  to  exercise  the  student. 

Ex.  1 . Find  the  equation  to  the  cissoid  of  Diodes  ; whose 
manner  of  description  is  as  below. 

From  any  two  points  p,  s,  at  equal 
distances  from  the  extremities  a,  b.  of 
the  diameter  of  a semicircle,  draw  st, 
pm,  perpendicular  to  ab.  From  the 
point  t where  st  cuts  the  semicircle, 

Iraw  a right  line  at,  it  will  cut  pm  in 
«,  a point  of  the  curve  required. 

Now,  by  theor.  87  Geom.  as  . sb  = st2  ; and  by  the  com 
truction,  as  . sb  = ap  . pb.  Also  the  similar  triangles  apm, 

f\sr,  give  ap  : pm  : : as  : st  : : pb  : st=PM— ~.  Conse- 

AP 

^ ol.  II,  38  quently 


290 


EQUATIONS  TO  CURVES. 


PM2 


PB2  pm2 

= ap.pb  and  lastly  - 


PB2 


= AP.AP2 


q'.eatly  st“  = Apl  . 

or  pa3  = pb  . pm2.  Hence  if  the  diameter  ab  = d,  ap  = x, 
pm  = y ; the  equation  is  x3  ~ y2  ( d — x). 

The  complete  cissoid  will  have  another  branch  equal  and 
similar  to  amq,  but  turned  contrary  ways  ; being  drawn  by 
means  of  points  t'  falling  in  the  other  half  of  the  circle.  But 
the  same  equation  will  comprehend  both  branches  of  the 
curve  ; because  the  square  of—  y,  as  well  as  that  of  -j-  y,  is 
positive. 

Cor.  All  cissoids  are  similar  figures  ; because  the  abscissae 
and  ordinates  of  several  cissoids  will  be  in  the  same  ratio, 
when  either  of  them  is  in  a given  ratio  to  the  diameter  of  its 
generating  circle. 

Ex.  2.  Find  the  equation  to  the  logarithmic  curve  whose 
fundamental  property  is,  that  when  the  abscissas  increase  or 
decrease  in  arithmetical  progression,  the  corresponding  ordi- 
nates increase  or  decrease  in  geometrical  progression. 

Ans.  y — ax,a  being  the  number  whose  logarithm  is  1,  in 
the  system  of  logarithms  represented  by  the  curve. 

Ex.  3.  Find  the  equation  to  the  curve  called  the  Witch , 
whose  construction  is  this  : a semicircle  whose  diameter  is  ab 
being  given  ; draw,  from  any  point  p in  the  diameter,  a per- 
pendicular ordinate,  cutting  the  semicircle  in  d,  and  terminat- 
ing in  m.  so  that  ap  : pd  : : ab  : hm  ; then  is  m always  a point 
in  the  curve.  Ans  = 


PROBLEM  IL 


Given  the  Equation  to  a Curve,  to  Describe  it,  and  trace  its 
Chief  Properties. 

The  method  of  effecting  this  is  obvious  : for  any  abscissas 
being  assumed,  the  corresponding  values  of  the  ordinates  be-  i 
come  known  from  the  equation  ; and  thus  the  curve  may  be 
traced,  and  its  limits  and  properties  developed. 

Ex.  1.  Let  the  equation  y3  = a2x,  or  y = \/a2x,  to  a line 
of  the  third  order  be  proposed. 

First,  drawing  the  two  indefinite  lines 
bh,  nc,  to  make  an  angle  bac  equal  to 
the  assumed  angle  of  the  co-ordinates  ; 
let  the  values  of  x be  taken  upon  ac, 
and  those  of  y upon  ab,  or  upon  lines 
parallel  to  ab.  Then,  let  it  be  enquired 
whether  the  curve  passes  through  the 
point  a,  or  not.  In  order  to  this,  we 
must  ascertain  what  y will  be  when 

x = t? 


EQUATIONS  TO  CURVES. 


291 


x = 0 : and  in  that  case  y = \/(a2  X0),that  is,  y — 0.  There- 
fore the  curve  passes  through  a.  Let  it  next  be  ascertained 
whether  the  curve  cuts  the  axis  ac  in  any  other  point  ; in  or- 
der to  which,  find  the  value  of  x when  y — 0 : this  will  be 
3/«a  x — 0,  or  x — 0.  Consequently  the  curve  does  not  cut 
the  axis  in  any  other  point  than  a.  Make  x = ap  = \a, 
and  the  given  equa  will  become  y = y ta]  = a%/h-  There- 
fore draw  pm  parallel  to  ab  and  equal  to  a\/\,  so  will  si  be 
a point  in  the  curve.  Again,  make  x — ac  = a ; then  the 
equation  will  give  y = ^/a3=  a.  Hence,  drawing  cn  parallel 
to  ab,  and  equal  to  ac  or  a,  n will  be  another  point  in  the 
curve.  And  by  assuming  other  values  of  y,  other  ordinates, 
and  consequently  other  points  of  the  curve,  may  be  obtained. 
Once  more,  making  x infinite,  or  x — cc,  we  shall  have  y = 
%/{a2  X co)  ; that  is.  y is  infinite  when  x is  so  ; and  therefore 
the  curve  passes  on  to  infinity.  And  further,  since  when  x 
is  taken  = 0,  it  is  also  y — 0,  and  when  x — cc,  it  is  also 
y — oo  ; the  curve  will  have  no  asymptotes  that  are  parallel 
to  the  co-ordinates. 

Let  the  right  line  an  be  drawn  to  cut  pm  (produced  if  ne- 
cessary) ins.  Then  because  cn  = ac,  it  will  be  ps=AP=ia. 
But  pm  = a3/!  = jal/4,  which  is  manifestly  greater  than  \a  ; 
so  that  pm  is  greater  than  ps,  and  consequently  the  curve  is 
concave  to  the  axis  ac 

Now.  because  in  the  given  equation  y3  — a2  r the  exponent 
©f  x is  odd,  when  x is  taken  negatively  or  on  the  other  side 
of  a,  its  sign  should  be  changed,  and  the  reduced  equation 
will  then  be  y — \/  — a2x.  Here  it  is  evident  that,  when  the 
values  of  r are  taken  in  the  negative  way  from  a towards  d, 
but  equal  to  those  already  taken  the  positive  way,  there  will 
result  as  many  negative  values  of  y,  to  fall  below  ad,  and  each 
equal  to  the  corresponding  values  of  y,  taken  above  ac. 
Hence  it  follows  that  the  branch  am'n'  will  bq  similar  and 
equal  to  the  branch  amn  ; but  contrarily  posited. 

Ex.  2.  Let  the  lemniscate  be  proposed,  which  is  a line  of 
the  fourth  order,  denoted  by  the  equation  a2y2  = a2x 2 — a:4 . 

I In  this  equation  we  have  y =±^-^/(a2  — x2)  ; 

where,  when  x = 0,  y = 0,  therefore  the  curve 
passes  through  a,  the  point  from  which  the  va- 
lues of  x are  measured.  When  r = ± a,  then 
y = 0 ; therefore  the  curve  passes  through  b 
and  c,  supposing  ab  and  ac  each  = ± a.  If  x 
were  assumed  greater  than  a,  the  value  of  y 
would  become  imaginary  ; therefore  no  part  of 
the  curve  lies  beyond  b or  c.  When  a;  = i-a. 


then 


§92 


EQUATIONS  TO  CURVES, 


then  y — \^/a*  -\a2  — \a^/ 3 ; which  is  the  value  of  the 
semi-ordinate  pw  when  ap  = \ ab.  And  thus,  by  assuming 
other  values  of  sr,  other  values  of  y may  be  ascertained,  and 
the  curve  described.  It  has  obviously  two  equal  and  similar 
parts,  and  a double  point  at  a.  A right  line  may  cot  this 
curve  in  either  2 points,  or  in  4 : even  the  right  line  bac  is 
conceived  to  cut  it  in  4 points  ; because  the  double  point  a is 
that  in  which  two  branches  of  the  curve,  viz  ma p,  and  tiaq, 
are  intersected. 


Ex.  3.  Let  there  be  proposed  the  Conchoid  of  the  ancients- 
which  is  a line  of  the  fourth  order  defined  by  the  equation 

(a2  —x-  . ( 'x  — 6)3  = x2y2 , or  y — ± 

Here,  if  x = 0,  then  y becomes  in- 
finite ; and  therefore  the  ordinate  at 
a (the  origin  of  the  abscissas)  is  an 
asymptote  to  the  curve.  It  ab  = 6, 
and  p be  taken  between  a and  b,  then 
shall  pm  and  pm  be  equal,  and  lie  on 
different  sides  of  the  abscissa  ap.  If 
x = b,  then  the  two  values  of  y vanish, 
because  x — 6 = 0,  and  consequently 
the  curve  passes  through  b,  having 
there  a double  point  if  ap  be  taken 
greater  than  ab,  then  will  there  be 
two  values  of  y,  as  before  having  contrary  signs  ; that  value, 
which  was  positive  before  being  now  negative,  and  vice  versa. 
But  if  ad  be  taken  = a,  and  p comes  to  d,  then  the  two  va- 
lues of  y vanish,  because  in  that  case  (a2  — x2)  = 0.  If  ap 
be  taken  greater  than  ad  or  a,  then  a 2 — x2  becomes  negative, 
and  the  value  of  y impossible  : so  that  the  curve  does  not  go 
beyond  d. 

Now  let  x be  considered  as  negative,  or  as  lying  on  the 

side  of  a towards  c.  Theny=  ± — — (a2 — x2).  Here 

if  x vanish,  both  these  values  of  y become  infinite  ; and  con- 
sequently the  curve  has  two  indefinite  arcs  on  each  side  the 
asymptote  or  directrix  ay.  If  .r  increase,  y manifestly  dimi- 
nishes ; and  when  x = a,  then  y vanishes  : that  is,  if  ac=ad, 
then  one  branch  of  the  curve  passes  through  c,  while  the 
other  passes  through  d.  Here  also,  if  x be  taken  greater 
than  a,  y becomes  imaginary  ; so  that  no  part  of  the  curve 
can  be  found  beyond  c. 

If  a = 6,  the  curve  will  have  a cusp  in  b,  the  node  between 
b and  d vanishing  in  that  case.  If  a be  less  than  6,  then  b 
will  become  a conjugate  point. 

In 


EQUATIONS  TO  CURVES. 


293 


In  the  figure,  m'cwi'  represents  what  is  termed  the  superior 
conekoid,  and  gbwidmbwi  the  inferior  conchoid.  The  point 
b is  called  the  pole  of  the  conchoid  ; and  the  curve  may  be 
readily  constructed  by  radial  lines  from  this  point,  by  means 

of  the  polar  equation  z = - ± a.  It  will  merely  be  re- 

quisite to  set  off  from  any  assumed  point  a,  the  distance 
ab  = b ; then  to  draw  through  b a right  line  ?wlm'  making 
any  angle  tp  with  cb,  and  from  l the  point,  where  this  line 
cuts  the  directrix  ay  (drawn  perpendicular  to  cb)  set  off  up- 
on it  cm'  = Lin  — a ; so  shall  m'  and  m be  points  in  the  supe- 
rior and  inferior  conchoids  respectively. 

Ex.  4 Let  the  principal  properties  of  the  curve  whose 
equation  is  yxn  = an  -j-  1 , be  sought  ; when  n is  an  odd  num- 
ber, and  when  n is  an  even  number. 

Ex.  5.  Describe  the  line  which  is  defined  by  the  equation 
xy  -)r  o-y  cy  = be  bx. 

Ex.  6.  Let  the  Cardioide,  whose  equation  is  y4 — Gay 3 -j- 
(2x2  + 12 a2)ys  — (6«x2  -f-  8a3)  y + (*2  + 3a2)  x2  — 0,  be 
proposed. 

Ex  7.  Let  the  Trident,  whose  equation  is  ary  = ax3  -j- 
bxa  4*  cx  -j-  d,  be  proposed. 

Ex.  8.  Ascertain  whether  the  Cissoid  and  the  Witch 
whose  equations  are  found  in  the  preceding  problem,  have 
asymptotes. 


PROBLEM  in. 


To  determine  the  Equation  to  any  proposed  Curve  surface. 

Here  the  required  equation  must  be  deduced  from  the  law 
or  manner  of  constructions  of  the  proposed  surface,  the  refer- 
ence being  to  three  co-ordinates,  commonly  rectangular  ones, 
the  variable  quantities  being  x,  y,  and  2.  Of  these,  two, 
namely,  x and  y,  will  be  found  in  one  plane,  and  the  third  2 
will  always  mark  the  distance  from  that  plane. 

Ex.  1.  Let  the  proposed  surface  be  that  of  a sphere,  fng. 

The  position  of  the  fixed  point  a, 
which  is  the  origin  of  the  co-ordinates 
ap,  pm,  mn,  being  arbitrary  ; let  it  be 
supposed,  for  the  greater  convenience, 
that  it  is  at  the  centre  of  the  sphere, 

Let  ma,  na,  be  drawn,  of  which  the 
latter  is  manifestly  equal  to  the  radius 
of  the  sphere,  and  may  be  denoted  by  r.  Then,  if,  ap  — x 
pm  =y,  mn  = 2;  the  right-angled  triangle  apm  will  give 


294 


EQUATIONS  TO  CURVE  SURFACES. 


am2  = ap2  + ™2  = x2  -{"  y2 • Id  like  manner,  the  right- 
angled  triangle  amn,  posited  in  a plane  perpendicular  to  the 
former,  will  give  an2  = am2  -f-  mn2,  that  is,  r-  = x2  + y2  -\-z 2 
or,  z2  — r2  — x2  —y2,  the  equation  to  the  spherical  surface, 
as  required 

Scholium.  Curve  surfaces,  as  well  as  plane  curves,  are 
arranged  in  orders  according  to  the  dimensions  of  the  equa- 
tions, by  which  they  are  represented.  And  in  order  to  de- 
termine the  properties  of  curve  surfaces,  processes  must  be 
employed,  similar  to  those  adopted  when  investigating  the 
properties  of  plane  curves.  Thus,  in  like  manner  as  in  the 
theory  of  curve  lines,  the  supposition  that  the  ordinate  y is 
equal  to  0,  gives  the  point  or  points  where  the  curve  cuts  its 
axis  ; so,  with  regard  to  curve  surfaces,  the  supposition  of 
z = 0,  will  give  the  equation  of  the  curve  made  by  the  in- 
tersection of  the  surface  and  its  base,  or  the  plane  of  the  co- 
ordinates x,  y.  Hence,  in  the  equation  to  the  spherical  sur- 
face, when  z — 0,  we  have  x2  + y2  — r2 , which  is  that  of  a 
circle  whose  radius  is  equal  to  that  of  the  sphere.  See  p.  534 
vol.  1. 

Ex.  2.  Let  the  curve  surface  proposed  be  that  produced 
by  a parabola  turning  about  its  axis. 

Here  the  abscissas  x being  reckoned  from  the  vertex  or 
summit  of  the  axis  and  on  a plane  passing  through  that  axis  ; 
the  two  other  co-ordinates  being,  as  before,  y and  z ; and 
the  parameter  of  the  generating  parabola  being  p the  equa- 
tion of  the  parabolic  surface  will  be  found  to  be  z2  + y2  — 
px  — 0. 

Now,  in  this  equation,  if  z be  supposed  = 0,  we  shall  have 
y2  =px,  which(pa.  534  vol.  1)  is  the  equation  to  the  generating 
parabola,  as  it  ought  to  be.  If  we  wished  to  know  what 
would  be  the  curve  resulting  from  a section  parallel  to  that 
which  coincides  with  the  axis,  and  at  the  distance  a from  it, 
we  must  put  z = a ; this  would  give  y2  = px  — a2 , which  is 
still  an  equation  to  a parabola,  but  in  which  the  origin  of  the 
abscissas  is  distant  from  the  vertex  before  assumed  by  the 
. a2 

quantity  — . 

Ex.  3.  Suppose  the  curve  surface  of  a right  cone  were 
proposed. 

Here  we  may  most  conveniently  refer  the  equation  of  the 
surface  to  the  plane  of  the  circular  base  of  the  cone.  In  this 
case,  the  perpendicular  distance  of  any  point  in  the  surface 
from  the  base,  will  be  to  the  axis  of  the  cone,  as  the  distance 
of  the  foot  of  that  perpendicular  from  the  circumference 

(measured 


EQUATIONS  TO  CURVE  SURFACES. 


295 


^measured  on  a radius),  to  the  radius  of  the  base  : that  is,  if 
the  values  of  x beeslimated  from  the  centre  of  the  base,  and 
r be  the  radius,  z will  vary  as  r — (.r2+y2).  Conse- 

quently, the  simplest  equation  of  the  conic  surface,  will  he 
z . r = — y/  (x3-f -y2),  or  r2  — 2 rz  -f-  z2  — - x2-\-y2. 

Now  from  this  the  nature  of  curves  formed  by  planes  cutting 
the  cone  in  different  directions,  may  readily  be  inferred.  Let 
it  be  supposed,  first,  that  the  cutting  plane  is  inclined  to  the 
base  of  a right-angled  cone  in  the  angle  of  45°,  and  passes 
through  its  centre  : then  will  z = x,  and  this  value  of  z sub- 
stituted for  it  in  the  equation  of  the  surface,  will  give  r2  — 
2 rx  — y2 , which  is  the  equation  of  the  projection  of  the  curve 
on  the  plane  of  the  cone’s  base  : and  this  (art.  3 of  this  chap.) 
is  manifestly  an  equation  to  a parabola. 

Or,  taking  the  thing  more  generally,  let  it  be  supposed  that 
the  cutting  plane  is  so  situated,  that  the  ratio  of  x to  z shall 
be  that  of  1 to  m : then  will  mx  — z,  and  m2  x2  — z2 . These 
substituted  for  z and  z 2 in  the  equation  of  the  surface,  will 
give,  for  the  equation  of  the  projection  of  the  section  on  the 
plane  of  the  base,  r2  — 2mx-\-(m2  — 1)  x2=y2 . Now  this 
equation,  if  m he  greater  than  unity,  or  if  the  cutting  plane 
pass  between  the  vertex  of  the  cone  and  the  parabolic  sec- 
tion, will  be  that  of  an  hyperbola  : and  if,  on  the  contrary, 
the  cutting  plane  pass  between  the  parabola  and  the  base,  i.  e. 
if  m be  less  than  unity,  the  term  (m2  — l):r3  will  be  negative, 
when  the  equation,  will  obviously  designate  any  ellipse. 

Schol.  It  might  here  be  demonstrated,  in  a nearly  simi- 
lar manner,  that  every  surface  formed  by  the  rotation  of  any 
conic  section  on  one  of  its  axes,  being  cut  by  any  plane  what- 
ever, will  always  give  a conic  section.  For  the  equation  of 
such  surface  will  not  contain  any  power  of  x,  y,  or  z,  greater 
than  the  second  ; and  therefore  the  substitution  of  any  values 
of  z in  terms  of  x or  of  y,  will  never  produce  any  powers  of 
x or  of  y exceeding  the  square.  The  section  therefore  must 
be  a line  of  the  second  order.  See,  on  this  subject,  Hutton’s 
Mensuration,  part  iii,  sect.  4. 

Ex.  3.  Let  the  equation  to  the  curve  surface  be  xyz  =*  a3. 

Then  will  the  curve  surface  bear  the  same  relation  to  the 
solid  right  angle,  which  the  curve  line  whose  equation  is 
xy  = a2  bears  to  the  plane  right  angle.  That  is,  the  curve 
surface  will  be  posited  between  the  three  rectangular  faces 
bounding  such  solid  right  angle,  in  the  same  manner  as  the 
equilateral  hyperbola  is  posited  between  its  rectangular  asymp- 
totes. And  in  like  manner  as  there  may  be  4 equal  equila- 
teral 


29  6 


CONSTRUCTION  OF  EQUATIONS. 


teral  hyperbolas  comprehended  between  the  same  rectangular 
asymptotes,  when  produced  both  ways  from  the  angular  point ; 
so  there  may  be  6 equal  hyperboloids  posited  within  the  6 
solid  right  angles  which  meet  at  the  same  summit,  and  all 
placed  between  the  same  three  asymptotic  planes. 


SECTION  II. 

On  the  Construction  of  Equations. 

PROBLEM  I. 


To  Construct  Simple  Equations,  Geometrically. 

Here  the  sole  art  consists  in  resolving  the  fractions,  to 
which  the  unknown  quantity  is  equal,  into  proportional  terms  ; 
and  then  constructing  the  respective  proportions,  by  means  oF 
probs.  8,  9,  10,  and  27  Geometry.  A few  simple  examples 
will  render  the  method  obvious. 

1.  Let  x = — ; then  c : a : : b : x.  Whence  x may  be 
found  by  constructing  according  to  prob.  9 Geometry. 

2.  Let  x — -f.  First  construct  the  proportion  d : a : : b : 

de 

^ , which  4th  term  call  g ; then  x = — ; or  e : c : : g : x. 

3.  Let  x — — . Then,  since  a2  — 62=(a+6)  X(a— b)  ; 

it  will  merely  be  necessary  to  construct  the  proportion 
c : a b ::  a — b : x. 


4.  Let  x 


as  b—bc- 
ad 


Find,  as  in  the  first  case,  g = ^-= 

d 


a2b  , , be  bes  he 

, and  li  = — , so  that  — may  = — 

ad  d ad  a 


first  case  = i — . 


So  shall  x 


Then  find  by  the 
g — i,  the  difference  of  those 


lines,  found  by  construction. 

5.  Let  x = a b~ First  find  the  fourth  proportional 


a/+Ac 

to  b,  a and  f,  which  make 


6 

h. 


Then  x = ; 

h-J-c 

or,  by  construction  it  will  be  h + c : a — d ::  a : x. 

2.  Let  a?  = ~ • .Make  the  right-angled  triangle  abc  such 

that 


CONSTRUCTION  OF  EQUATIONS. 


297 


that  the  leg  ab  = a,  bc  = b ; then  ac  — y/  (ab2 
-f*  bc2)  = y/  (a2+62),  by  th.  34  Geom.  Hence 

x — . Construct  therefore  the  proportion 

c : ac  : : ac  : x,  and  the  unknown  quantity  will 
be  found,  as  required. 

7.  Let  x = a—  First,  find  cd  a 

h-j-c 

mean  proportional  between  ac  — c,  and 
cb  = d,  that  is,  find  cd  = cd.  Then 
make  ce  = a,  and  join  de,  which  will 
evidently  be  = y/  (a2  -j-  cd).  Next  on 
any  line  eg  set  off  ef  = h .+  c,  eg  = ed  ; and  draw  gh 
parallel  to  fd,  to  meet  de  (produced  if  need  be)  in  h.  So 
shall  eh  be  = x,  the  third  proportional  to  h c,  and 
y/  (a2  -f-  cd),  as  required. 

Note  Other  methods  suitable  to  different  cases  which 
may  arise  are  left  to  the  student’s  invention.  And  in  all 
constructions  the  accuracy  of  the  results,  will  increase  with 
the  size  of  the  diagrams  ; within  convenient  limits  for 
operation. 

PROBLEM  II. 

To  Find  the  Roots  of  Quadratic  Equations  by  Construction. 

In  most  of  the  methods  commonly 
given  for  the  construction  of  quadratics, 
it  is  required  to  set  off  the  square  root 
of  the  last  term  ; an  operation  which 
can  only  be  performed  accurately  when 
that  term  is  a rational  square.  We  shall 
here  describe  a method  which,  at  the 
same  time  that  it  is  very  simple  in  prac- 
tice, has  the  advantage  of  showing  clearly 
the  relations  of  the  roots,  and  of  dividing  the  third  term  into 
two  factors,  one  of  which  as  least  may  be  a whole  number 

In  order  to  this  construction,  all  quadratics  may  be  classed 
under  4 forms  : viz. 

1.  x2  -f-  ax  — be  — 0. 

2.  x 2 — ax — be—  0. 

3.  x2  -f-  ax  -f-  be  — 0. 

, 4.  x2 — ax  -f-  6c  = 0. 

1.  One  general  mode  of  construction  will  include  the  first 
two  of  these  forms.  Let  x2  z p ar  — be  — 0.  and  b greater 
than  c.  Describe  any  circle  abp  having  its  diameter  not  less 
'han  the  given  quantities  a and  b — c,  and  within  this  circle 

V<”~  H.  39  inscribe 


AC  E a 


298  CONSTRUCTION  OF  EQUATIONS. 

inscribe  two  chords  ab  — a,  ad  = b — c,  both  from  any 
common  assumed  point  a.  Then  produce  ad  to  f so  that 
df  = c,  and  about  the  centre  c of  the  former  circle,  with  the 
radius  cf,  describe  another  circle,  cutting  the  chords  ad,  ab, 
produced  in  f,  e,  g,  h : so  shall  ag  be  the  affirmative  and 
ah  the  negative  root  of  the  equation  x2-\-ax  — be  = 0 ; and 
contrariwise  ag  will  be  the  negative  and  ah  the  affirmative 
root  of  the  equation  x2  — ax  - — be  — 0. 

For,  af  or  ad  -f  df  = 6,  and  df  or  ae  = c ; and,  making 
ag  or  bh  = x,  we  shall  have  ah  = a + x : and  by  the  pro- 
perty of  the  circle  egfh  (theor.  "61  Geom ) the  rectangle 
ea  . af  = ga  . ah,  or  be  = (a  -f-  x ) x,  or  again  by  transpo- 
sition x2  -\-ax  — be  = 0.  Also  if  ah  be  = — x,  we  shall  have 
ag  or  bh  or  ah  — ab  = — x — a : and  conseq.  ga  . ah  = 
x2  + ax,  as  before.  So  that,  whether  ag  be  = x,  or  ah  = 
— x,  we  shall  always  have  x2  + ax  — be  = 0.  And  by  an 
exactly  similar  process  it  may  be  proved  that  ag  is  the  nega- 
tive, and  ah  the  positive  root  of  x2 — ax — be— 0. 

Cor.  In  quadratics  * of  the  form  x2  + ax  — be  = 0,  the 
positive  root  is  always  less  than  the  negative  root  ; and  in  those 
of  the  form  x2  — ax  — be  = 0,  the  positive  root  is  always 
greater  than  the  negative  one. 

2.  The  third  and  fourth  cases  also  are 
comprehended  under  one  method  of  con- 
struction, with  two  concentric  circles.  Let 
x2  if  ax  -f  be  — 0.  Here  describe  any 
circle  abd,  whose  diameter  is  not  less  than 
either  of  the  given  quantities  a and  b -}-  c ; 
and  within  that  circle  inscribe  two  chords 
ab  = a,  ad  = b + c,  both  from  the  same 
point  a.  Then  in  ad  assume  df  = c,  and  about  c the  centre 
of  the  circle  abd.  with  the  radius  cf  describe  a circle,  cutting 
the  chords  ad,  ab,  in  the  points  f,  e,  g,  h : so  shall  ag,  ah, 
be  the  two  positive  roots  of  the  equation  x2  — ajr  + be  = 0, 
and  the  two  negative  roots  of  the  equation  x3  + ax  -j-  be  = 0. 
The  demonstration  of  this  also  is  similar  to  that  of  the  first 
case. 

Cor.  1.  If  the  circle  whose  radius  is  cf  just  touches  the 
chord  ab,  the  quadratic  will  have  two  equal  roots  which  can 
only  happen  when  }a2  = be. 

Cor.  2.  If  that  circle  neither  cut  nor  touch  the  chord  ah, 
the  roots  of  the  equation  will  be  imaginary  ; and  this  wil 
always  happen,  in  these  two  forms,  when  be  is  greater 
than  . 


PROBLEM 


CONSTRUCTION  OF  eUBICS,  &c. 


299 


PROBLEM  III. 

To  Find  the  Roots  of  Cubic  and  Biquadratic  Equations,  by 
Construction. 

1.  In  finding  the  roots  of  any  equation,  containing  only 
one  unknown  quantity,  by  construction,  the  contrivance  con- 
sists chiefly  in  bringing  a new  unknown  quantity  into  that 
equation  ; so  that  various  equations  may  be  had,  each  con- 
taining the  two  unknown  quantities  ; and  further,  such  that 
any  two  of  them  contain  together  all  the  known  quantities 
of  the  proposed  equation.  Then  from  among  these  equations 
two  of  the  most  simple  are  selected,  and  their  corresponding 
loci  constructed  ; the  intersection  of  those  loci  will  give  the 
roots  sought. 

Thus  i will  be  found  that  cubics  may  be  constructed  by 
two  parabolas,  or  by  a circle  and  a parabola,  or  byr  a circle 
and  an  equilateral  hyperbola,  or  by  a circle  and  an  ellipse, 
&c.  : and  biquadratics  by  a circle  and  a parabola,  or  by  a circle 
and  an  ellipse,  or  by  a circle  and  an  hyperbola,  &c  Now, 
since  a parabola  of  given  parameter  may  be  easily  constructed 
by  the  rule  in  cor.  2 th  4 Parabola,  we  selec:  the  circle  and 
the  parabola,  for  the  construction  of  both  biquadratic  and  cu- 
bic equations.  The  general  method  applicable  to  both,  will 
be  evident  from  the  following  description. 

2.  Let  m"  am'm  be  a parabola  whose 
axis  is  ap,  m"  m'gm  a circle  whose  cen- 
tre is  c and  radius  cm,  cutting  the  pa- 
rabola, in  the  points  m,  m',  m",  m " : 
from  these  points  draw  the  ordinates 
to  the  axis  mp,  mV,  m''p",  m'''p"  : and 
from  c let  fall  cd  perpendicularly  to 
the  axis  : also  draw  cn  parallel  to  the 
axis  : meeting  pm  in  n.  Let  ad  = a, 

-dc  = b,  cm  = n,  the  parameter  of  the 
parabola  = p,  at  = x , pm  = y.  Then  (pa.  534  vol.  1 )px  — y2 : 
also  cm2  = cn2  -f  nm2,  orn2  = (x  ^ a)2  -f-  (y:p  b )2  ; that 
is,  x2  2 ax  a2  -f-  y2  ± 2 by  + b2  — n2 . Substituting  in 

this  equation  for  *,  its  value~p  and  arranging  the  terms  ac 

cording  to  the  dimensions  of  y,  there  will  arise 

y4  dt  (2 pa  + p2)y2  ± 2 bp2y  + (a2  -f-  b2  — n2)p2  — 0, 
a biquadratic  equation  whose  roots  will  be  expressed  by  the 
ordinates  pm,  f'm',  p"m'',  p'"m  ",  at  the  points  of  intersection 
of  the  given  parabola  and  circle. 

3 To  make  this  coincide  with  any  proposed  biquadratic 
wjhose  second  term  is  taken  away  (by  cor.  theor.  3)  ; assume 

y4„ 


A 


300 


CONSTRUCTION  OF  CUBICS,  kc. 


y*  — qy2  + ry  — s = 0.  Assume  also  p = 1 ; then  eom- 
paringthe  terms  of  the  two  equations,  it  will  be,  2 a — 1 = q, 

or  a = = r>  or  & ==— - ; a2  + 62  — re2  = — s,  or 

n2  — a2  b2  s,  and  consequently  n — (a2  -f  b-  + s). 

Therefore  describe  a parabola  whose  parameter  is  1,  and  in 

n 1 m 

the  axis  take  ad  = — — : at  right  angles  to  it  draw  dc  and 

— — ir  ; from  the  centre  c,  with  the  radius  (a2 -ft2  + s), 
describe  the  circle  m'mgm,  cutting  the  parabola  in  the  points 
m,  m',  m ',  m*''  ; then  the  ordinates  pm,  p'm',  p m",  p"'m"‘,  | 
will  be  the  roots  required. 

Note.  This  method,  of  making  p = 1,  has  the  obvious 
advantage  of  requiring  only  one  parabola  for  any  number  of 
biquadratics,  the  necessary  variation  being  made  in  the  radius 
of  the  circle. 


Cor.  1.  When  dc  represents  a negative  quantity,  the  ordi- 
nates on  the  same  side  of  the  axis  with  c represent  the  nega- 
tive roots  of  the  equation  ; and  the  contrary. 

Cor.  2.  If  the  circle  touch  the  parabola,  two  roots  of  the 
equation  are  equal  ; if  it  cut  it  only  in  two  points,  or  touch  it 
in  one,  two  roots  are  impossible  ; and  if  the  circle  fall  wholly 
within  the  parabola,  all  the  roots  are  impossible. 


Cor.  3.  If  a2  -f-  l2  = n2 , or  the  circle  pass  through  the 
point  a,  the  last  term  of  the  equation,  i.  e.  a2  + l2  —n2)p2  =0  ; 
and  therefore  y4  ± (2 pa  p2)y-  ± 2 hp2y  = 0,  or 

y 3 ± (2 pa  + p2)y  ± 2 bp2  — 0.  This  cubic  equation  may 
be  made  to  coincide  with  any  proposed  cubic,  wanting  its 
second  term,  and  the  ordinates  pm,  p'  m",  p'  m'",  are  its  roots. 

Thus,  if  the  cubic  be  expressed  generally  by  ys±qy±s=0. 
By  comparing  the  terms  of  this  and  the  preceding  equation, 
we  shall  have  :±  2 pa  + p2  — ± q,  and  ± 2 bp2  — ± s,  or 

zz.  a — ip  qi  and  b = So  that,  to  construct  a 

cubic  equation,  with  any  given  parabola,  whose  half  parameter 
is  ab  (see  the  preceding  figure)  ; from  thq  point  b take  in 
the  axis,  (forward  if  the  equation  have  — q , but  backward  if 


q be  positive)  the  line  bd  ; then  raise  the  perpendicular 


dc  = — , and  from  c describe  a circle  passing  through  the 
2 ^ ^ 
vertex  a of  the  parabola ; the  ordinates  pm,  kc  drawn  from 
the  points  of  intersection  of  the  circle  and  parabola,  will  be 
the  roots  required. 

PROBLEM 


CONSTRUCTION  OF  CUBICS. 


301 


PROBLEM  IV. 

To  Construct  an  Equation  of  any  Order  by  means  of  a Locus 

of  the  same  Degree  as  the  Equation  proposed,  and  a Right 

Line. 

As  the  general  method  is 
the  same  in  all  equations,  let 
it  be  one  of  the  5th  degree,  as 
x5  - bxi-\-acx3  — a 2 dx 2 -f -a3 ex 
— o4/= 0.  Let  the  last  term 
a4/  be  transposed  ; and,  tak- 
ing one  of  the  linear  divisors, 
f,  of  the  last  term,  make  it 
equal  to  z,  for  example,  and  divide  the  equation  by  a 4 ; then 
...  xs — bx*  4- acxs —a^dxi astx 

Will  2 = . 

<J4 

On  the  indefinite  line  bq  describe  the  curve  of  this  equa- 
tion, bmdrlfc,  by  the  method  taught  in  prob.  2,  sect.  1,  of 
this  chapter,  taking  the  values  of  x from  the  fixed  point  b. 
The  ordinates  pm,  sr,  &c.  will  be  equal  to  2 ; and  therefore, 
from  the  point  b draw  the  right  line  ba  = /,  parallel  to  the 
ordinates  pm,  sr,  and  through  the  point  a draw  the  inde- 
finite right  line  kc  both  ways,  and  parallel  to  bq.  From  the 
points  in  which  it  cuts  the  curve,  let  fall  the  perpendiculars, 
mp,  rs,  cq  : they  will  determine  the  abscissas  bp,  bs,  bq, 
which  are  the  roots  of  the  equation  proposed.  Those  from 
a towards  a are  positive,  and  those  lying  the  contrary  way  are 
negative. 

If  the  right  line  ac  touch  the  curve  in  any  point,  the  cor- 
responding abscissa  x will  denote  two  equal  roots  ; and  if  it 
do  not  meet  the  curve  at  all,  all  the  roots  will  be  imaginary. 

If  the  sign  of  the  last  term,  a4/,  had  been  positive,  then 
we  must  have  made  z = — f,  and  therefore  must  have  taken 
ba  = — /,  that  is,  below  the  point  p,  or  on  the  negative  side. 

Exercises. 

Ex.  1.  Let  it  be  proposed  to  divide  a given  arc  of  a circle 
into  three  equal  parts. 

Suppose  the  radius  of  the  circle  to  be  represented  by  r, 
the  sine  of  the  given  arc  by  a,  the  unknown  sine  of  its  third 
part  by  x,  and  let  the  known  arc,  be  3 u,  and  of  course,  the 
required  arc  be  u.  Then,  by  equa.  vm,  ix,  chap,  iii,  we 
shall  have 

„ * / ~ , \ sin  2u  ■ cos  «4- cos  2u  • sin  u 

sin  3 u — sin  (2m  -+-  u)  ~ — — — — , 

. _ . , , % 2 sin  it . cos  u 

sin  2m  = sin  ( m -f-  m)  = , 

cos  2u  =ss  cos  ( u + «)  =: 


Putting 


302 


TRISECTION  OP  AN  ARCH. 


Putting,  in  the  fir^t  of  these  equations,  for  sin  3u  its  given 
value  a,  and  for  sin  2 u,  cos  2m,  their  values  given  in  the  two  . 
other  equations,  there  will  arise 

3 mo  u . eos2  u . sin2  u 

a — . 

r 

Then  substituting  for  sin  m its  value  x,  and  for  cos2  u its 
value  r2  — x 2 and  arranging  all  the  terms  according  to  the 
powers  of  x,  we  shall  have 

X3 — %r2x-\-\  ar-  =0, 

a cubic  equation  of  the  forma:3  — px  -f*  q = 0,  with  the 
condition  that  2VP3  > l92  ; that  is  to  say,  it  is  a cubic  equa- 
tion falling  under  the  irreducible  case,  and  its  three  root*  are 
represented  by  the  sines  of  the  three  arcs  u,  u -}-  120°,  and 
v -f-  240°. 

Now,  this  cubic  may  evidently  be  constructed  by  the  rule 
in  prob.  3 cor.  3 But  the  trisection  of  an  arc  may  also  be 
effected  by  means  of  an  equilateral  hyperbola,  in  the  following 
manner. 


Let  the  arc  to  be  trisected  be  ab. 

In  the  circle  abc  draw  the  semi- 
diameter  ad,  and  to  ad  as  a diame- 
ter. and  to  the  vertex  a,  draw  the 
equilateral  hyperbola  ae  to  which 
the  right  line  ab  (the  chord  of  the 
arc  to  be  trisected)  shall  be  a tangent  in  the  point  a ; then 
the  arc  af,  included  within  this  hyperbola,  is  one  third  of  the 


arc  AB. 

For,  draw  the  chord  of  the  arc  ae,  bisect  ad  at  g,  so  that 
G will  be  the  centre  of  the  hyperbola,  join  df,  and  draw  gh 
parallel  to  it,  cutting  the  chords  ab,  af,  in  i and  k Then, 
the  hyperbola  being  equilateral,  or  having  its  transverse  and 
conjugate  equal  to  one  another,  it  follows  from  Def.  16  Conic 
Sections,  that  every  diameter  is  equal  to  its  parameter,  and 
from  cor.  theor.  2 Hyperbola,  that  gk  ki  = ak2,  or  that 
gk  : ak  : : ak  : in  ; therefore  the  triangles  gka.  aki  are 
similar,  and  the  angle  kai  = agk,  which  is  manifestly  ='  adf. 
Now  the  angle  adf  at  the  centre  of  the  circle  being  equal  to 
3ai  or  fab  ; and  the  former  angle'  at  the  centre  being  mea- 
sured by  the  arc  af,  while  the  latter  at  the  circumference  is 
measured  by  half  fb  ; it  follows  that  af  = 4fb,  or  = i ab,  as 
it  ought  to  be. 

Ex.  2.  Given  the  side  of  a cube,  to  find  the  side  of  another 
of  double  capacity. 

Let  the  side  of  the  given  cube  be  a,  and  that  of  a double 
one  y,  then  2 a3  =y 3 , or  by  putting  2a  = b,  it  will  be  a-=by3  ; 
there  are  therefore  to  be  found  two  mean  proportionals  be- 
tween 


DUPLICATION  OF  THE  CUBE. 


303 


tween  the  side  of  the  cube  and  twice  that  side,  and  the  first 
of  those  mean  proportionals  will  be  the  side  of  the  double 
cube.  Now  these  may  be  readily  found  by  means  of  two  pa- 
rabolas ; thus : 

Let  the  right  lines  ar,  as,  be  joined 
at  right  angles  ; and  a parabola  amh  be 
described  about  the  axis  ar,  with  the 
parameter  a ; and  another  parabola  ami 
about  the  axis  as,  with  the  parameter  6 : £ 
cutting  the  former  in  m.  Then  ap  = x, 
pm  = y , are  the  two  mean  proportionals 
of  which  y is  the  side  of  the  double  cube  required. 

For,  in  the  parabola  amh  the  equation  is  y2  = ax,  and  in 
the  parabola  ami  it  is  x2  = by.  Consequently  a : y : : y : x, 
and  y : x : : x : b.  Whence  yx  = ab  ; or,  by  substitution, 
y y/ by  — ab,  or  by  squaring  y3b  ==a2b2  ; or  lastly,  ys  =,  a2 b 
==  2a3,  as  it  ought  to  be. 


THE 


[ 304  j 


THE  DOCTRINE  OF  FLUXIONS. 

DEFINITIONS  AND  PRINCIPLES 


Art.  i.  In  the  Doctrine  of  Fluxions,  magnitudes  or  quar. 
tities  of  all  kinds  are  considered,  not  as  made  up  of  a number 
of  small  parts,  but  as  generated  by  continued  motion,  by 
means  of  which  they  increase  or  decrease.  .4s,  a line  by 
the  motion  of  a point  ; a surface  by  the  motion  of  a line  ; 
and  a solid  by  the  motion  of  a surface.  So  likewise,  time 
may  be  considered  as  represented  by  a line,  increasing  uni- 
formly by  the  motion  of  a point.  And  quantities  of  all  kinds 
whatever,  which  are  capable  of  increase  and  decrease,  may  in 
like  manner  be  represented  by  geometrical  magnitudes,  con- 
ceived to  be  generated  by  motion. 

2.  Any  quantity  thus  generated,  and  variable,  is  called  a 
Fluent,  or  a Flowing  Quantity.  And  the  rate  or  proportion 
according  to  which  any  flowing  quantity  increases,  at  any 
position  or  instant,  is  the  Fluxion  of  the  said  quantity,  at  that 
position  or  instant  : and  it  is  proportional  to  the  magnitude 
by  which  the  flowing  quantity  would  be  uniformly  increased 
in  a given  time,  with  the  generating  celerity  uniformly  con- 
tinued during  that  time. 

3.  The  small  quantities  that  are  actually  generated,  pro- 
duced, or  described,  in  any  small  given  time,  and  by  any 
continued  motion  either  uniform  or  variable,  are  called  In- 
crements. 

4.  Hence,  if  the  motion  of  increase  be  uniform,  by  which 
increments  are  generated,  the  increments  will  in  that  case  be 
proportional,  or  equal,  to  the  measures  of  the  fluxions  : but 
if  the  motion  of  increase  be  accelerated,  the  increment  so 
generated,  in  a given  finite  time,  will  exceed  the  fluxion : 
and  if  it  be  a decreasing  motion,  the  increment,  so  generated, 
will  be  less  than  the  fluxion.  But  if  the  time  be  indefinitely 
small,  so  that  the  motion  be  considered  as  uniform  for  that 
instant  ; then  these  nascent  increments  will  always  be  pro- 
portional, or  equal,  to  the  fluxions,  and  may  be  substituted  in- 
stead of  them,  in  any  calculation 


5.  To 


FLUXIONS. 


305 


5.  To  illustrate  these  definitions  : Sup- 
pose a point  m be  conceived  to  move  from  

the  position  a,  and  to  generate  a line  ap, 

by  a motion  any  how  regulated  ; and 
suppose  the  celerity  of  the  point  in,  at 
any  position  p,  to  be  such,  as  would,  if  from  thence  it  should 
become  or  continue  uniform,  be  sufficient  to  cause  the  point 
to  describe,  or  pass  uniformly  over,  the  distance  p p,  in  the 
given  time  allowed  for  the  fluxion  : then  will  the  said  line  p p 
represent  the  fluxion  of  the  fluent,  or  flowing  line,  ap,  at  that 
position. 

6.  Again,  suppose  the  right 

line  mn  to  move,  from  the  posi- 
tion ab,  continually  parallel  to 
itself,  with  any  continued  motion, 
so  as  to  generate  the  fluent  or 
flowing  rectangle  abqp,  while  the 
point  m describes  the  line  ap  : also,  let  the  distance'  p p be 
taken,  as  before,  to  express  the  fluxion  of  the  line  or  base 
ap  ; and  complete  the  rectangle  p Qqp.  Then,  like  as  p p 

is  the  fluxion  of  the  line  ap,  so  is  p q,  the  fluxion  of  the  flowing 
parallelogram  aq.  : both  these  fluxions,  or  increments,  being 
uniformly  described  in  the  same  time. 

7.  In  like  manner,  if  the  solid 
aerp  be  conceived  to  be  gene- 
rated by  the  plane  pqr,  moving 
from  the  position  abe,  always 
parallel  to  itself,  along  the  line 
ad  ; and  if  p p denote  the  fluxion 
of  the  line  ap  : Then,  like  as  the 
rectangle  p <iqp,  or  pq  X vp,  de- 
notes the  fluxion  of  the  flowing  rectangle  ab^p,  so  also  shall 
the  fluxion  of  the  variable  solid,  or  prism  aber^p,  be  de- 
noted by  the  prism  pqr rqp,  or  the  plane  pr  X p p.  And,  in 
both  the  last  two  cases,  it  appears  that  the  fluxion  of  the 
generated  rectangle,  or  prism,  is  equal  to  the  product  of  the 
generating  line,  or  plane  drawn  into  the  fluxion  of  the  line 
along  which  it  moves. 

8.  Hitherto  the  generating  line,  or  plane,  has  been  con- 
sidered as  of  a constant  and  invariable  magnitude  ; in  which 
case  the  fluent,  or  quantity  generated,  is  a rectangle,  or  a 
prism,  the  former  being  described  by  the  motion  of  a line, 
and  the  latter  by  the  motion  of  a plane.  So,  in  like  manner 
are  other  figures,  whether  plane  or  solid,  conceived  to  be  de- 

Vo  l.  II.  40  scribed 


E R r 


l\ 

\ 

\ 

c 

Ir 

Q 

9 

M_ 

\ 

\ 

A P P D 


B 


o T a 


n 


F P D 


306  DEFINITIONS. 

scribed  by  the  motion  of  a Variable  Magnitude,  whether  it 
be  a line  or  a plane.  Thus.  let  a variable  line  pq  be  carried 
by  a parallel  motion  along  ap  ; or  while  a point  p is  carried 
along,  and  describes  the  line  ap,  suppose  another  point,. 


q to  be  carried  by  a motion  perpendicular  to  the  former  j 
and  to  describe  the  line  pq  : let  pq  be  another  position  r 
of  pq,  indefinitely  near  to  the  former ; and  draw  Qr  pa-  j 
rallel  to  ap  Now  in  this  case  there  are  several  fluents,  j 
or  flowing  quantities,  with  their  respective  fluxions  : name-  l 
ly,  the  line  or  fluent  ap,  the  fluxion  of  which  is  p p or  Qr  ; d 
the  line  or  fluent  pq,  the  fluxion  of  which  is  rq  ; the  curve  i 
or  oblique  fine  aq,  described  by  the  oblique  motion  of  the 
point  q,  the  fluxion  of  which  is  q.q  ; and  lastly,  the  sur- 
face apq,  described  by  the  variable  line  pq,  the  fluxion  of  i 
which  is  the  rectangle  f Qrp,  or  pq  X pjo.  In  the  same  manner  j 
may  any  solid  be  conceived  to  be  described,  by  the  motion  of 
a variable  plane  parallel  to  itself,  substituting  the  variable  i 
plane  for  the  variable  line  ; in  which  case  the  fluxion  of  the  I 
solid,  at  any  position,  is  represented  by  the  variable  plane,  at 
that  position,  drawn  into  the  fluxion  of  the  line  along  which  it 
is  carried. 

9.  Hence  then  it  follows  in  general,  that  the  fluxion  of 
any  figure,  whether  plane  or  solid,  at  any  position,  is  equal 
to  the  section  of  it,  at  that  position,  drawn  into  the  fluxion  of 
the  axis,  or  line  along  which  the  variable  section  is  sup- 
posed to  be  perpendicularly  carried  : that  is,  the  fluxion  of 
the  figure  aqp,  is  equal  to  the  plane  pq  X f p,  when  that 
figure  is  a solid,  or  to  the  ordinate  pq  X f p,  when  the  figure  is 
a surface. 

10.  It  also  follows  from  the  same  premises,  that  in  any 
curve  or  oblique  line  aq,  whose  absciss  is  ap,  and  ordinate  is 
pq,  the  fluxions  of  these  three  form  a small  right-angled 
plane  triangle  q qr  ; for  qj*  = rp  is  the  fluxion  of  the  absciss 
ap,  qr  the  fluxion  of  the  Grdinate  pq,  and  q q the  fluxion  of 
the  curve  or  right  line  aq  And  consequently  that,  in  any 
curve,  the  square  of  the  fluxion  of  the  curve,  is  equal  to  the 

sum 


NOTATION. 


307 


sum  of  the  squares  of  the  fluxions  of  the  absciss  and  ordinate, 
when  these  two  are  at  right  angles  to  each  other. 

11.  From  the  premise*  it  also  appears,  that  contemporane- 
ous flueuts,  or  quantities  that  flow  or  increase  together,  which 
are  always  in  a constant  ratio  to  each  other,  have  their 
fluxions  also  in  the  same  constant  .ratio,  at  every  position, 
For.  let  ap  and  bq.  be  two  contempo- 
raneous fluents,  described  in  the  same  ^ ^ ’ ’ ' ’ P 

time  by  the  motion  of  the  points  p and 
q,  the  contemporaneous  positions  be-  ~ r>  V 

ing  p,  q,  and  p,  q ; and  let  ap  be  to 
bq,  or  Ap  to  b^,  constantly  in  the  ratio 
of  1 to  n.  i 

Then is  » X ap  = Bq, 

and  it  X .ip  = b q ; 
therefore,by  subtraction,  n X p p = qy; 
that  is,  the  fluxion  • pp  : fluxion  qy  : : 1 : n, 
the  same  as  the  fluent  ap  : fluent  Bq  : : 1 : n, 
or,  the  fluxions  and  fluents  are  in  the  same  constant  ratio. 

But  if  the  ratio  of  the  fluents  be  variable,  so  will  that  of 
the  fluxions  be  also,  though  not  in  the  same  variable  ratio 
with  the  former,  at  every  position. 


NOTATION,  kc. 

12.  To  apply  the  foregoing  principles  to  the  determination 
•f  the  fluxions  of  algebraic  quantities,  by  means  of  which 
those  of  all  other  kinds  are  assigned,  it  will  be  necessary  first 
to  premise  the  notation  commonly  used  in  this  science,  with 
some  observations.  As,  first,  that  the  final  letters  of  the 
alphabet  z,  y,  x,  u,  &c.  are  used  to  denote  variable  or  flow- 
ing quantities  ; and  the  initial  letters,  a,  b,  c,  d,  &c.  to  denote 
constant  or  invariable  ones  : Thus,  the  variable  base  ap  of 
the  flowing  rectangular  figure  abqp,  in  art.  6,  may  be  repre- 
sented by  x ; and  the  invariable  altitude  pq,  by  a : also,  the 
variable  base  or  absciss  ap,  of  the  figures  in  art.  8,  may  be 
represented  by  x,  the  variable  ordinate  pq,  by  y • and  the 
variable  curve  or  line  Aq,  by  z. 

Secondly,  that  the  fluxion  of  a quantity  denoted  by  a 
single  letter,  is  represented  by  the  same  letter  with  a point 
over  it : Thus,  the  fluxion  of  x is  expressed  by  x,  the  fluxion 
of  y by  y,  and  the  fluxion  of  z by  i.  As  to  the  fluxions  of 
constant  or  invariable  quantities,  as  of  a,  b,  c,  &c.  they  are 
equal  to  nothing,  because  they  do  not  flow  or  change  their 
magnitude. 


Thirdly, 


308 


DIRECT  METHOD  OF  FLUXIONS. 


Thirdly,  that  the  increments  of  variable  or  flowing  quan- 
tities, are  also  denoted  by  the  same  letlt  rs  with  a small ' over 
them  : Thus,  the  increments  of  x,  y,  z , aie  x , y,  z . 

13.  From  these  notations,  and  the  foregoing  principles,  the 
quantities  and  their  fluxions,  there  consideied,  will  be  denoted 
as  below.  Thus,  in  all  the  foregoing  figures,  put 

the  variable  or  flowing  line  - - ap  = x, 

in  art  6,  the  constant  line  - - pq  = a, 

in  art  8 the  variable  ordinate  - pq  = y, 

also,  the  variable  line  or  curve  - ah  — z : 

Then  shall  the  several  fluxions  be  thus  represented,  namely, 
x — ep  the  fluxion  of  the  line  ap, 
a.x  — PHqp  the  fluxion  of  abop  in  art.  6, 
yx  — Pqrp  the  fluxion  of  apo.  in  art.  8, 
z ~ Qq  — x/  (x2  + y2)  the  fluxion  of  aq.  ; and 
ax  — p r the  fluxion  of  the  solid  in  art.  7,  if  a denote  the 
constant  generating  plane  pur  ; also, 
nx  = BQ  in  the  figure  to  art.  11,  and 
nx  = Q q the  fluxion  of  the  same. 

14.  The  principles  and  notation  being  now  laid  down,  we 
may  proceed  to  the  practice  and  rules  of  this  doctrine  ; which 
consists  of  two  principal  parts,  called  the  Direct  and  Inverse 
Method  of  Fluxions  ; namely,  the  direct  method,  which 
consists  m finding  the  fluxion  of  any  proposed  fluent  or 
flowing  quantity  ; and  the  inverse  method,  which  consists  in 
finding  the  fluent  of  any  proposed  fluxion.  As  to  the  former 
of  these  two  problems,  it  can  always  be  determined,  and  that 
in  finite  algebraic  terms  ; but  the  latter,  or  finding  of  fluents, 
can  only  be  effected  in  some  certain  cases,  except  by  means 
of  infinite  series. — First  then,  of 

THE  DIRECT  METHOD  OF  FLUXIONS. 

To  fnd  the  Fluxion  of  the  Product,  or  Rectangle  of  txzo  Porta- 
ble Quantities. 

15.  Let  arqp,  x y,  be  the  flow- 

ing or  variable  rectangle,  generated 
by  two  hues  Pfc  and  rq.,  moving  al- 
ways perpendicular  to  each  other, 
from  the  positions  ar  and  ap  ; deno- 
ting the  one  by  x and  the  other  by  y ; 
supposing  x and  y to  be  so  related, 
that  the  curve  line  a^  may  always 
pass  through  the  intersection  q.  of  those  lines,  or  the  opposite 
augle  of  the  rectangle.  Now, 


DIRECT  METHOD  OF  FLUXIONS. 


309 


Now,  the  rectangle  consists  of  the  two  trilinear  spaces 
ap®,  arq,  of  which,  the 

fluxion  of  the  former  is  pq  X p p,  or  yx, 
that  of  the  latter  is  - rq  X ar,  or  x'y , by  art.  8 ; 
therefore  the  sum  of  the  two  xy  + x'y,  is  the  fluxion  of  the 
whole  rectangle  xy  or  arqp. 


The  Same  Otherwise. 


16.  Let  the  sides  of  the  rectangle  x and  y,  by  flowing, 
become  x -f-  x'  and  y + y : then  the  product  of  these  two,  or 
xy  + xy'  + yx'  -{-  x'y'  will  be  the  new  or  contemporaneous 
value  of  the  flowiug  rectangle  pr  or  xy  : subtract  the  one 
value  from  the  other,  and  the  remainder,  xy  yx'-\~  x'y  , will 
be  the  increment  generated  in  the  same  time  as  x'  or  y ; of 
which  the  last  term  xy  is  nothing  or  indefinitely  small,  in  res- 
pect of  the  other  two  terms,  because  x'  and  y are  indefinitely 
small  in  respect  of  x and  y ; which  term  being  therefore  omit- 
ted, there  remains  xy  -f-  yx'  for  the  value  of  the  increment  ; 
and  hence,  by  substituting  x and  y , for  x‘  and  y,  to  which  they 
are  proportional,  there  arises  xy  -f-  yx  for  the  true  value  of 
the  fluxion  of  xy  ; the  same  as  before. 


17.  Hence  may  be  easily  derived  the  fluxion  of  the 
powers  and  products  of  any  number  of  flowing  or  variable 
quantities  whatever  ; as  of  xyz,  or  uxyz,  or  vuxyz,  &c.  And 
flrst,  for  the  fluxion  of  xyz  : putp  = xy,  and  the  whole 
given  fluent  xyz  — q,  or  q = xyz  = pz'.  Then,  taking  the 
'fluxions  of  q = pz,  by  the  last  article,  they  are  q = pz  -j- 
pz ; but  p — xy,  and  so  p = xy  4*  xy  by  the  same  article  : 
substituting  therefore  these  values  of  p and  p instead  of 
them,  in  the  value  of  q,  this  becomes  q — xyz  + x'yz  + xy'z , 
the  fluxion  of  xyz  required  ; w’hich  is  therefore  equal  to  the 
sum  of  the  products,  arising  from  the  fluxion  of  each  letter, 
or  quantity,  multiplied  by  the  product  of  the  other  two. 

Again,  to  determine  the  fluxion  of  uxyz,  the  continual 
product  of  four  variable  quantities  ; put  this  product,  namely 
uxyz,  or  qu  = r,  where  q = xyz  as  above.  Then,  taking  the 
fluxions  by  the  last  article,  r — qu  q'u;  which,  by  sub- 
stituting for  q and  q their  values  as  above,  becomes 
r = uxyz  -f-  uxyz  + uxyz  + uxyz,  the  fluxion  of  uxyz  as 
required  : consisting  of  the  fluxion  of  each  quantity,  drawn 
into  the  products  of  the  other  three. 

In 


310 


DIRECT  METHOD  OF  FLUXIONS. 


In  the  very  same  manner  it  is  found,  that  the  fluxion  of 
vuxyz  is  v uxyz  -f  vuxyz  -f  vuxyz  -f  vuxyz  -f  vuxyz ; and 
so  on,  for  any  number  of  quantities  whatever  ; in  which  it 
is  always  found,  that  there  are  as  many  terms  as  there  are 
variable  quantities  iu  the  proposed  fluent  ; and  that  these 
terms  consist  of  tbe  fluxioa  of  each  variable  quantity,  mul- 
tiplied by  the  product  of  all  the  rest  of  the  quantities. 

18.  Hence  is  easily  derived  the  fluxion  of  any  power  of 
a variable  quantity,  as  of  x2,  or  x3,  or  .r4,  &c.  For,  in  the 
product  or  rectangle  xy,  if  x — y,  then  is  xy  «=  n or  x2, 
and  also  its  fluxion  xy  -f  x'y  = xx  -f  x'x  or  2xjr,  the  fluxion 
of  x2 . 

Again,  if  all  the  three  .r,  y,  z be  equal  ; then  is  the  product 
of  the  three  xyz  — x3  ; and  consequently  its  fluxion  xyz  -f 
xyz-{-xyz  = xxx-j-xxx  -f  xxx  or  3x2.r,  the  fluxion  of  x3. 

In  the  same  manner,  it  will  appear  that 
the  fluxion  of  x4  is  = 4x3,z.  and 
the  fluxion  of  x5  is  = 5x*x  and,  in  general, 
the  fluxion  of  x”  is  = nxn~lx  ; 
where  n is  any  positive  whole  number  whatever. 

That  is,  the  fluxion  of  any  positive  integral  power  is  equal 
to  the  fluxion  of  the  root  (x),  multiplied  by  the  exponent 
of  the  power  (w),  and  by  the  power  of  the  same  root  whose 
index  is  less  by  1,  (xn_1). 


And  thus,  the  fluxion  of  a -f  cx  being  cx, 


that  of  (a  -f  cx) 
that  of  (a  -fcx2)2 
that  of  (x2  -f  y2  )2 
that  of  (x  -fey2)3 


s 2 cx  X (a  + cx)  or  2aci- -f  2c2  xx, 
s 4cxiX  (a  -f  cx" ) or  4 acxx  -f  4c2x3x, 
s (4xx  + 4yy)  X (xJ  + y2), 
s (3f  +6cyy)X(x  -fey3)2. 


19.  From  the  conclusions  in  the  same  article,  we  may 
also  derive  the  fluxion  of  any  fraction,  or  the  quotient  of  one 
variable  quantity  divided  by  another,  as  of 

-.  For,  put  the  quotient  or  fraction  - = q ; then,  multiplying 

by  the  denominator,  x — qy  ; and,  taking  the  fluxions, 

X = yy  + qy,  or  qy  = x — q’y  ; and,  by  division, 

q — - — — = (by  substituting  the  value  of  q,  or'^), 

^ =?-— the  fluxion  of  -,  as  required. 

y y 2 V2  V ^ 


That 


DIRECT  METHOD  OF  FLUXIONS. 


311 


That  is  the  fluxion  of  any  fraction,  is  equal  to  the  fluxion 
of  the  numerator  drawn  into  the  denominator,  minus  the 
fluxion  of  the  denominator  drawn  into  the-  numerator,  and 
the  remainder  divided  by  the  square  of  the  denominator. 

So  that  the  fluxion  of  — is  a X or  axy~  . 

y y 2 _ y 2 

20.  Henc  e loo  is  easily  derived  the  fluxion  of  any  negative 

integer  power  of  a variable  quantity,  as  of  or-”,  or  which 

is  the  same  thing.  For  here  the  numerator  of  the  fraction 
is  1,  whose  fluxion  is  nothing  ; and  therefore,  by  the  last 
article,  the  fluxion  of  such  a fraction,  or  negative  power, 
is  barely  equal  to  minus  the  fluxion  of  the  denominator, 
divided  by  the  square  of  the  said  denominator.  That  is  the 

a „ l - nxn~ >x  tlx  . 

fluxion  of  x~n,  or  — is  — or -r-  or  — nx~n—}  x ; 

xn  x2  n Xn  1 

or  the  fluxion  of  any  negative  integer  power  of  a variable 
quantity  as  x~n,  is  equal  to  the  fluxion  of  the  root,  multiplied 
by  the  exponent  of  the  power,  and  by  the  next  power  less 
by  I ; the  same  rule  as  for  positive  powers. 

The  same  thing  is  otherwise  obtained  thus  : Put  the 

proposed  fraction,  or  quotient  ~ = q ; then  is  qxn  = 1 ; 

and,  taking  the  fluxions,  we  have 

qxn  -f  qnxn~lx=  0,  hence  qxTi=—qnxn~'lx  ; divide  by  xn,  then 

k = - ^ = (by  substituting  -1  for  q ),  or  = — 

nX-n-i  x . the  same  as  before. 

Hence  the  fluxion  of  x -1  or  — is  — x~2x  or , 

x xs 

1 2x 

that  of  * x~2  or  — is— Zx-*  x or , 

x-  x 

1 . 3i 

that  of  - x~3  or  — r is— 3x~*x  or 

x 3 x 45 

, , 4ci.y4 

ax-*  or—  is  — 4aa:-s.r  or , 


that  of 
that  of  (a+x)-1  or 
that  of  c(a4-3x3)~2or 


a+  x 

c 


is  — (a- \-x)~2x  or 


(a  + x)3’ 


(a+ 3x2)z 


is — 12 cxx  X (a-}-3x2)-3  9 


or 


12  cix 


(a+ 3*2)3’ 

21.  Much  in  the  same  manner  is  obtained  the  fluxion  of 

m 

any  fractional  power  of  a fluent  quantity,  as  of  xn , or  %/  xm. 

m 

For,  put  the  proposed  quantity  xn  = q ; then,  raising  each 
side  te  the  n power,  gives  xm—  qn  ; 

taking 


312 


DIRECT  METHOD  OF  FLUXIONS, 


taking  the  fluxions,  gives  mxm~vx  = nqn~ ' q ; then 

mxm~'x  mxm~xx  „ m , 
dividing  by  nq""1,  gives  q = no«-i  = m=lT  * » 

nxm~-n 

Which  is  still  the  same  rule,  as  before,  for  finding  the  fluxion 
of  any  power  of  a fluent  quantity,  and  which  therefore  is  gen- 
eral, whether  the  exponent  be  positive  or  negative,  integral 

XL  x * 

or  fractional.  And  hence  the  fluxion  of  ax  2 is  ax2x, 

dec  dec 

1_  L X . — - - 

that  of  ax2  is  ^ax2~1x—^«,x~2x  = 2^/x  ; and  that  of 

i i . — 

y/  {a?  — x2')  or  (a2  — x2')2  is  ^(a2  — x2)2  X — 2xx  — 

v/  (n-  — >r2  ) 

22.  Having  norv  found  out  the  fluxions  of  all  the  ordi- 
nary forms  of  algebraical  quantities  ; it  remains  to  deter- 
mine those  of  logarithmic  expressions  and  also  of  exponential 
ones,  that  is  such  powers  as  have  their  exponents  variable  or 
flowing  quantities.  And  first,  for  the  fluxion  of  Napier’s,  oi 
the  hyperbolic  logarithm. 

23.  Now,  to  determine  this  from 
the  nature  of  the  hyperbolic  spaces. 

Let  a be  the  principle  vertex  of  an 
hyperbola,  having  its  asymptotes  cd, 
cp,  with  the  ordinates  da,  ba,  pq, 

&c  parallel  to  them.  Then,  from 
the  nature  of  the  hyperbola  and  of 

logarithms,  it  is  known,  that  any  space  abp$  is  the  log.  of 
the  ratio  of  cb  to  cp,  to  the  modulus  abcd.  Now,  put 
1 = cb  or  ba  the  side  of  the  square  or  rhombus  db  ; 
m = the  modulus,  or  cb  X ba  ; or  area  of  db,  or  sine  of 
the  angle  c to  the  radius  1 ; also  the  absciss  cp  = r,  and 
the  ordinate  pq.  = y.  Then,  by  the  nature  of  the  hyperbola, 
cp  X pci  is  always  equal  to  db,  that  is,  xy  — m : hence 

v = -,  and  the  fluxion  of  the  space,  xy  is  — = p Qjp  the 

J X x 

fluxion  of  the  log.  of  x,  to  the  modulus  m.  And,  in 
the  hyperbolic  logarithms,  the  modulus  in  being  1,  there- 
fore - is  the  fluxion  of  the  hyp.  log.  of  x ; which  is  therefore 

equal  to  the  fluxion  of  the  quantity,  divided  by  the  quantity 
itself. 

Hence  the  fluxion  of  the  hyp.  log. 

of  l + xisih: 

of  1 — x is  - — — , 

1 —x 


of 


OF  SECOND,  THIRD,  &c.  FLUXIONS, 


313 


of  x 4-  z is 

X + Z 

x(a  — x)  Fj(a-i-*) 


-a-fx  . 

of is 

a — x 

of  axn  is 


(a  — xy 


X 


a - x 2a'x 
a-j-x  a3— x 2’ 


24.  By  means  of  the  fluxions  of  logarithms,  are  usually 
determined  those  of  exponential  quantities,  that  is,  quan- 
tities which  have  their  exponent  a flowing  or  variable  letter. 
These  exponentials  are  of  two  kinds,  namely,  when  the  root 
is  a constant  quantity,  as  ex,  and  when  the  root  is  variable  as 
well  as  the  exponent,  as  yT. 


25.  In  the  first  case  put  the  exponential,  whose  fluxion 
is  to  be  found,  equal  to  a single  variable  quantity  2,  namely, 
z — e x then  take  the  logarithm  of  each,  so  shall  log,  z~-xX 

log.  e ; take  the  fluxions  of  these,  so  shall  - = i X log.  e, 

by  the  last  article  : hence  s — z'x  X log.  e — e*  x X log.  e, 
which  is  the  fluxion  of  the  proposed  quantity  c*  or  2 ; and 
which  therefore  is  equal  to  the  said  given  quantity  drawn  into 
the  fluxion  of  the  exponent,  and  into  the  log.  of  the  root. 

Hence  also,  the  fluxion  of  (a-j-c)nj:  is  (a-f-c)"*  Xn'x  X log. 
(a+c). 

26.  In  like  manner,  in  the  second  case,  put  the  given 
quantity  yx  = 2 ; then  the  logarithms  give  log.  2 = x X log.  y, 

and  the  fluxions  give  -"  = j X log.  y + xX  y-  ; hence 

z — zx  X log.  y -f-  — - = (by  substituting  yx  for  2)  X 

log.  y + xy*~vy,  which  is  the  fluxion  of  the  proposed  quan- 
tity yx  ; and  which  therefore  consists  of  two  terms,  of  which 
the  one  is  the  fluxion  of  the  given  quantity  considering  the 
exponent  as  constant,  and  the  other  the  fluxion  of  the  same 
quantity  considering  the  root  as  constant. 


OF  SECOND,  THIRD,  &c.  FLUXIONS. 


Having  explained  the  manner  of  considering  and  determin- 
ing the  first  fluxions  of  flowing  or  variable  quantities  ; it  re* 
Vbr„,  If,  41  mains 


314 


OF  SECOND,  THIRD,  kc.  FLUXIONS. 


mains  now  to  consider  those  of  the  higher  orders,  as  second, 
third,  fourth,  &c.  fluxions. 

27.  If  the  rate  or  celerity  with  which  any  flowing  quantity 
changes  its  magnitude,  be  constant,  or  the  same  at  every  posi- 
tion ; then  is  the  fluxion  of  it  also  constantly  the  same.  But  I 
if  the  variation  of  magnitude  be  continually  changing,  either 
increasing  or  decreasing  ; then  will  there  be  a certain  degree 
of  fluxion  peculiar  to  every  point  or  position  ; and  the  rate  of 
variation  or  change  in  the  fluxion,  is  called  the  Fluxion  of  the 
Fluxion,  or  the  Second  Fluxion  of  the  given  fluent  quantity. 

In  like  manner,  the  variation  or  fluxion  of  this  second  fluxion, 
is  called  the  Third  Fluxion  of  the  first  proposed  fluent  quan-  < 
tity  ; and  so  on. 

These  orders  of  fluxions  are  denoted  by  the  same  fluent 
letter  with  the  corresponding  number  of  points  over  it  ; 
namely,  two  points  for  the  second  fluxion,  three  points  for 
the  third  fluxion,  four  points  for  the  fourth  fluxion,  and  so  on.  i 
So,  the  different  orders  of  the  fluxion  of  x,  are  x,  x,  i,  x > &c. , 
where  each  is  the  fluxion  of  the  oue  next  before  it. 

28.  This  description  of  the  higher  orders  of  fluxions 
may  be  illustrated  by  the  figures  exhibited  in  art.  8,  page  3U6  : j. 
where,  if  x denote  the  absciss  ap,  and  y the  ordinate  fq.  : ; 
and  if  the  ordiuate  p$  or  y flow  along  the  absciss  ap  or  x , 
with  a uniform  motion  ; then  the  fluxion  of  x,  namely, 
x = vp  or  Qr,  is  a constant  quantity,  or  x = 0,  in  all  the 
figures.  Also,  in  fig.  1,  in  which  aq.  is  a right  line,  y = rq, 
or  the  fluxion  of  pq,  is  a constant  quantity,  or  y = 0 ; for, 
the  angle  q,  — the  angle  a,  being  constant,  or  is  to  rq,  or 
x to  y , in  a constant  ratio.  But  in  the  2d  fig.  rq,  or  the 
fluxion  of  pq,  continually  increases  more  and  more  ; and 
in  fig.  3 it  continually  decreases  more  and  more,  and  there- 
fore in  both  these  cases  y has  a second  fluxion,  being  positive 
in  fig.  2,  hut  negative  in  fig.  3.  And  so  on,  for  the  other  or- 
ders of  fluxions. 

Thus  if,  for  instance,  the  nature  of  the  curve  be  such, 
that  x 3 is  every  where  equal  to  o2y  ; then,  taking  the  fluxions 
it  is  a2y  = Ba2x  ; and,  considering  x always  as  a constant 
quantity,  and  taking  always  the  fluxions,  the  equations  of 
the  several  orders  of  fluxions  will  be  as  below,  viz. 

the  1st  fluxions  a2'y  — 3x2x, 
the  2d  fluxions  a 2y  = Gxx2 , 
the  3d  fluxions  a2:''.  = 6^3, 
the  4tb  fluxions  a2  “ — 0, 

aud  all  the  higner  fluxions  also  = 0,  or  nothing. 

Alse 


OF  SECOND  THIRD,  &c.  FLUXIONS. 

Also,  the  higher  orders  of  fluxions  are  found  in  the  same 
manner  as  the  lower  ones.  . Thus, 
the  first  fluxion  of  y3  is  - - - - 3 y2y  ; 

its  2d  flux,  or  the  flux,  of  3 y2y,  con-  } 
sidered  as  the  rectangle  of  3y2,}3y2y  -j- 6yy  2 : 

and  y , is ) 

and  the  flux,  of  this  again,  or  the  3d  ) _ „ . . , . ...  . „. 
flux,  of  y3,  is  - IWy  + 'Zyyy+Sy’- 


. 29.  In  the  foregoing  articles,  it  has  been  supposed  that 
the  fluents  increase,  or  that  their  fluxions  are  positive  ; but 
it  often  happens  that  some  fluents  decrease,  and  that  there- 
fore their  fluxions  are  negative  : and  whenever  this  is  the 
case,  the  sign  of  the  fluxion  must  be  changed,  or  made  con- 
trary to  that  of  the  fluent.  So,  of  the  rectangle  xy,  when 
both  x and  y increase  together,  the  fluxion  is  xy  + x‘y  ; but 
if  one  of  them,  as  y,  decrease,  while  the  other,  x,  increases  ; 
then,  the  fluxion  of  y being  — y , the  fluxion  of  xy 
will  in  that  case  be  x y — x'y  . This  may 
be  illustrated  by  the  annexed  rectangle, 
apqr  = xy,  supposed  to  be  generated 
by  the  motion  of  the  line  pq  from  a to-  R 
wards  c,  abd  by  the  motion  of  the  line 
Rq  from  b towards  a : For,  by  the  mo- 
tion of  pq,  from  a towards  c,  the  rect- 
angle is  increased,  and  its  fluxion  is  -f- 
j xy  ; but,  by  the  motion  of  rq,  from  b to-  A PC 

wards  a,  the  rectangle  is  decreased,  and 
the  fluxion  of  the  decrease  is  x'y  ; there- 
fore, taking  the  fluxion  of  the  decrease  from  that  of  the  in- 
crease, the  fluxion  of  the  rectangle  xy,  when  x increases  and 
: y decreases,  is  xy  — xy. 


cc 

0 

y y 

X 

! 

| 

REMARK  BY  THE  EDITOR. 

The  fluxion  of  the  algebraic  quantity  xy  is  properly  yx  + xy 
in  all  cases  of  increase  or  decrease.  We  should  always  use 
the  signs  of  the  fluxions  of  algebraic  expressions  as  those 
signs  arise  from  the  known  rules,  without  considering  whether 
the  quantities  increase  or  decrease  ; but  in  denoting,  algebrai- 
cally, the  simple  fluxions  of  geometrical  quantities,  we  should 
prefix  the  sign  minus  to  the  fluxions  of  such  as  decrease  : 
and  thus  we  may,  in  any  case,  use  the  fluxions  of  algebraic 
equations,  together  with  the  fluxions  derived  from  geometrical 
figures,  without  embarrassment  or  apprehension  of  error. 

30.  We 


316 


RULES  FOR  FINDING 


30.  We  may  now  collect  all  the  rules  together,  which  have 
been  demonstrated  in  the  foregoing  arti;les,  for  finding  the 
fluxions  of  all  sorts  of  quantities.  And  hence, 

1st,  For  the  fluxion  of  any  Power  of  a flowing  quantity. 
— Multiply'  all  together  the  exponent  of  the  power,  the  flux- 
ion of  the  root,  and  the  power  next  less  by  1 of  the  same 
root. 

£d,  For  the  fluxion  of  the  Rectangle  of  two  quantities.  —Mul- 
tiply each  quantity  by  the  fluxion  of  the  other,  and  connect 
the  two  products  together  by  their  proper  signs. 

3d.  For  the  fluxion  of  the  Continual  product  of  any  number 
of  flowing  quantities , — Multiply  the  fluxion  of  each  quantity 
by  the  product  of  all  the  other  quantities,  and  connect  all 
the  products  together  by  their  proper  signs. 

4th,  For  the  fluxion  of  a Fraction. — From  the  fluxion  of  the 
numerator  drawn  into  the  denominator,  subtract  the  fluxion 
of  the  denominator  drawn  into  the  numerator,  and  divide  the 
result  by  the  square  of  the  denominator. 

5th,  Or,  the  2d,  3d,  and  4th  cases  may  be  all  winded  undei 
one,  and' performed  thus. — Take  the  fluxion  of  the  given  ex- 
pression as  often  as  (h  re  are  variable  quantities  in  it  sup- 
posing first  only  or  e of  them  variable,  and  the  rest  constant  : 
then  another  \ amble,  and  the  rest  constant  ; and  so  on, 
till  they  have  all  in  their  turns  been  singly  supposed  variable, 
and  connect  all  these  fluxions  together  with  their  own  signs. 

6th.  For  the  fluxion  of  a Logarithm. — Divide  the  fluxion 
of  the  quantity  by  the  quantity  itself,  and  multiply  the  result 
by  the  modulus  of  the  system  of  logarithms. 

Note.  The  modulus  of  the  hyperbolic  logarithms  is  1, 
and  the  modulus  of  the  common  logs,  is  . 0-4342944S. 

7th,  For  the  fluxion  of  an  Exponential  quantity  having  the 
Root  Constant. — Multiply  altogether  the  given  quantity  the 
fluxion  of  its  exponent,  and  the  the  hyp.  log.  of  the  root. 

8th.  For  the  fluxion  of  an  Exponential  quantity  having  ike 
Root  Variable. — To  the  fluxion  of  the  given  quantity,  found 
by  the  1st  rule,  as  if  the  root  only  were  variable,  and  the 
fluxion  of  the  same  quantity  found  by  the  7th  rule,  as  if  the 
exponent  only  were  variable  ; and  the  sum  will  be  the  fluxion 
for  both  of  them  variable. 

Note.  When  the  given  quantity  consists  of  several  terms, 
find  the  fluxion  of  each  term  separately,  and  connect  them 
all  together  with  their  proper  signs. 


SI.  PRACTICAL 


FLUXIONS. 


317 


51.  PRACTICAL  EXAMPLES  TO  EXERCISE  THE  FOREGOING 
RULES. 

1.  The  fluxion  of  axy  is 

2.  The  fluxion  of  bxyz  is 

3.  The  fluxion  of  cx  X ( ax — cy)  is 

4.  The  fluxion  of  xmyn  is 

5.  The  fluxion  of  xmyn zr  is 

6.  The,  fluxion  of  (x  y)  X ( x~y ) is 

7.  The  fluxion  of  2 ax3  is 

8.  The  fluxion  of  2x3  is 

9.  The  fluxion  of  3 x*y  is 

10.  The  fluxion  of  4 x3y*  is 

-1 

11.  The  fluxion  of  ax2y  — x2y3  is 

12.  The  fluxion  of  4xi—x2y  -f-  3 byz  is 

_I 

13.  The  fluxion  of  x on"  is 

m 

24.  The  fluxion  of  !J/ xm  or  x~^  is 

15.  The  fluxion  of — 1 — or  - ora;—”1  is 

jy  xm  ™ n 

xn 

x 

16.  The  fluxion  of  y/  x or  x-  is 

_1 

17.  The  fluxion  of  %/  x or  x3  is 

18.  The  fluxion  of  %/x 2 or  x 3 is 

3. 

19.  The  fluxion  of  y/x3  or  x2  is 

3. 

20.  The  fluxion  of  */x3  or  x 4 is 

4. 

21.  The  fluxion  of  %/x 4 or  x3  is 

22.  The  fluxion  of  y/  ( a 2 + x2)  or  (a2  + x2)2  is 

23.  The  fluxion  of  (a2  — x2)  or  (a2  — x2')2  is 

24.  The  fluxion  of  y/  (2 rx  — xx ) or  (2 rx  — xx)2  is 

25.  The  fluxion  of - or  (a2  — x2)  2 is 

v/(a2-x2)  v J 

26.  The  fluxion  of  ( ax — xx)3  is 


27.  The 


3 IS 


FINDING  OF  FLUXIONS. 


27.  The  fluxion  of  2x  y/ a 2 ± x2  is 

28.  The  fluxion  of  (a2  — x2)2  is 

29.  The  fluxion  of  y/xz , or  ( xz y is 

30.  The  fluxion  of  y/xz-zz  or  (xz  — zz)2  is 

31.  The  fluxion  of  — or  — -x2  is 

a^x  3 

32.  The  fluxion  of  is 

a + x 

ocrri 

33.  The  fluxion  of  — is 

yn 

34.  The  fluxion  of  — is 

z 

35.  The  fluxion  of  — is 

XX 

36.  The  fluxion  of  is 

a—x 

37.  The  fluxion  of  z.  is 

x-f  z 

38.  The  fluxion  of  is 

z* 

2 

39.  The  fluxion  of  — is 

Si  0 

y 2 

40.  The  fluxion  of  a—^—  is 

z 

41.  The  fluxion  of  is 

42.  The  fluxion  of  the  hyp.  log.  of  ax  is 

43.  The  fluxion  of  the  hyp.  log.  of  1 + is 

44.  The  fluxion  of  the  hyp.  log-  of  1 — x is 

45.  The  fluxion  of  the  hyp.  log.  of  x 2 is 

46.  The  fluxion  of  the  hyp.  log.  of  y/  z is 

47.  The  fluxion  of  the  hyp.  log.  of  xm  is 


48.  The 


FINDING  OF  FLUENTS. 


319 


48.  The  fluxion  of  the  hyp  log.  of 


2 . 
- 19 

X 


49.  The  fluxion  of  the  hyp.  log.  of 

50.  The  fluxion  of  the  hyp.  log.  of 


1+  x . 

18 

1 — x 
1- 


l-\-  X 


IS 


51.  The  fluxion  of  c*is 

52.  The  fluxion  of  10*  is 

53.  The  fluxion  of  (a  + c)*  is 

54.  The  fluxion  of  100*!' is 

55.  The  fluxion  of  xz  is 

56.  The  fluxion  of  yXox  is 

57.  The  fluxion  of  xx  is 

58.  The  fluxion  of  (xy)xs  is 

59.  The  fluxion  of  xy  is 

60.  The  fluxion  of  x y"  is 

61.  The  second  fluxion  of  xy  is 

62.  The  second  fluxion  of  xy,  when  x is  constant,  is 

63.  The  second  fluxion  of  xn  is  ‘ 

64.  The  third  fluxion  of  xn,  when  x is  constant,  is 

65.  The  third  fluxion  of  xy  is 


THE  INVERSE  METHOD,  OR  THE  FINDING  OF 
FLUENTS. 

32.  It  has  been  observed,  that  a Fluent,  or  Flowing  Quan- 
tity, is  the  variable  quantity  which  is  considered  as  increasing 
or  decreasing.  Or,  the  fluent  of  a given  fluxion,  is  such  a 
quantity,  that  its  fluxion,  found  according  to  the  foregoing 
rules,  shall  be  the  same  as  the  fluxion  given  or  proposed. 

33.  It  may  further  be  observed,  that. Contemporary  Fluents, 
or  Contemporary  Fluxions,  are  such  as  flow  together,  or  for 
the  same  time. — When  contemporary  fluents  are  always 
equal,  or  in  any  constant  ratio  ; then  also  are  their  fluxions 
respectively  either  equal,  or  in  that  same  constant  ratio. 
That  is,  if  x — y,  then  is  x — y ; or  if  x : y : : n : 1,  then  is 
x : y : : n : 1 ; or  if  x =■  ny,  then  is  x — ny  . 

34.  It  is  easy  to  find  the  fluxions  to  all  the  given  forms  of 
fluents  ; but,  on  the  contrary,  it  is  difficult  to  find  the  fluents 

: of  many  given  fluxions  : and  indeed  there  are  numberless 

cases 


320 


FINDING  OF  FLUENTS. 


eases  in  which  this  cannot  at  all  be  done,  excepting  by  the 
quadrature  and  rectification  of  curve  lines,  or  by  logarithms, 
or  by  infinite  series.  For,  it  is  only  in  certain  particular  forms 
and  cases  that  the  fluents  of  given  fluxions  can  be  found  ; 
there  being  no  method  of  performing  this  universally,  a priori, 
by  a direct  investigation,  like  finding  the  fluxion  of  a given 
fluent  quantity.  We  can  only  therefore  lay  down  a few 
rules  for  such  forms  of  fluxions  as  we  know,  from  the  direct 
method,  belong  to  such  and  such  kinds  of  flowing  quantities  : 
and  these  rules,  it  is  evident,  must  chiefly  consist  in  perform- 
ing such  operations  as  are  the  reverse  of  those  by  which  the 
fluxions  are  found  of  given  fluent  quantities.  The  principal 
cases  of  which  are  as  follow. 

35.  To  find  the  Fluent  of  a Simple  Fluxion  ; or  of  that  in  which 

there  is  no  variable  quantity , and  only  one  jluxional  quantity. 

This  - is  done  by  barely  substituting  the  variable  or  flowing 
quantity  instead  of  its  fluxion  ; being  the  result  or  reverse  of 
the  notation  only. — Thus, 

The  fluent  of  afi  is  ax. 

The  fluent  of  ay  + 2y  is  ay  -f-  2y. 

The  fluent  of  a2  + x2  is  a-  + x2. 

36.  When  any  Power  of  a flowing  quantity  is  Multiplied  by 

the  Fluxion  of  the  Root ; 

Then,  having  substituted,  as  before,  the  flowing  quantity 
for  its  fluxion,  divide  the  result  by  the  new  index  of  the 
power.  Or,  which  is  the  same  thing,  take  out  or  divide  by. 
the  fluxion  of  the  root : add  1 to  the  index  of  the  power  ; 
and  divide  by  the  index  so  increased.  Which  is  the  reverse 
of  the  1st  rule  for  finding  fluxions. 


So,  if  the  fluxion  proposed  be  - - 3x5x 

Leave  out,  or  divide  by,  x then  it  is  - 3xs  ; 

add  1 to  the  index,  and  it  is  - - 3x-6  ; 

divide  by  the  index  6,  and  it  is  - - |x6  or  -i-x6. 


which  is  the  fluent  of  the  proposed  fluxion  3xsf. 

In  like  manner, 

The  fluent  of  2ax-j^  is  ax. 

The  fluent  of  3x%'x  is  x3. 

\ The 


FINDING  OF  FLUENTS. 


321 


1_  . 3. 

The  fluent  of  4x2  x is  |x2* 

The  fluent  of  2t/7y  is  -f y4- 

The  fluent  of  az^z  is  TeTaz  6 . 

The  fluent  of  x^-r-f-  3 y3y  is  §x^+ 1 yz  • 
The  fluent  of  xn-1.r  is  jxn. 

The  fluent  of  wyn-ly  is 

The  fluent  of  — or  z-2z  is 
z- 

The  fluent  of -{7-  is 

y' 

The  fluent  of  (a  -f-  x)*x  is 
The  fluent  of  («4  + y4)y3y  is 
The  fluent  of  (a3  -f-  z3)4z2z  is 
The  fluent  of  (an  -f-  xn)taxn-lx'  is 
The  fluent  of  (a1  -j-  y2)3yy  is 

The  fluent  of  7 is 

*/(«2+z2) 

The  fluent  of  -A — r is 

v/(  a—x) 


37.  When  the  Root  under  a Vinculum  is  a Compound  Quantity  ; 
and  the  Index  of  the  part  or  factor  Without  the  Vinculum , in- 
creased by  1,  is  some  Multiple  of  that  under  the  Vinculum  : 

Put  a single  variable  letter  for  the  compound  root  ; and  sub- 
stitute its  powers  and  duxion  instead  of  those  of  the  same  value, 
in  the  given  quantity  ; so  will  it  be  reduced  to  a simpler  form, 
to  which  the  preceding  rule  can  then  be  applied. 

. 2. 

Thus,  if  the  given  fluxion  be  f = (a2-j-x2)3x3  *,  where  3, 
the  index  of  the  quantity  without  the  vinculum,  increased 
by  1,  making  4,  which  is  just  the  double  of  2,  the  exponent 
of  x2  within  the  vinculum  : therefore,  putting z = a2- (-  x3, 
thence  x2  — z — a2, the  fluxion  of  which  is  2 x’x—'z;  hence 
then  x3  x = \x2  z — \z  ( z—a2),  and  the  given  fluxion  j,  or 

(a2  +x2)3x3i,  is  = \z  3 z (z  - a2),  or=4z3  z — |a3z3  j ; and 

8.  5_  “ 5_ 

hence  the  fluent  f is  = T3gZ3  — T\a2z'3  = 3 z3  (tl  z — T3?a2  ). 
Or,  by  substituting  the  value  of  z instead  of  it,  the  same 

rV*2  -TV»2),°rfV 

42 


f a 2 +x2)X(x2  - fa2). 


fluent is3(a2 +x2 ) 3 X( 


322 


FINDING  OF  FLUENTS 


In  like  manner  for  the  following  example*. 

To  find  the  fluent  of  a + cx  X x3x . 

3_ 

To  find  the  fluent  of  (a  -f-  cx)4  x2x. 

To  find  the  fluent  of  (a  -j-  cx2)3  X dx3  x- 

To  find  the  fluent  of  — or  (a~Fz')2czz  ■ 

' F a -f-j  z 

3n— l * — 1 

To  find  the  fluent  of  or(a-| -za)2cz3n~1zJ 

s/  +«“ 

To  find  the  fluent  of-  v ai  or  (a 2 -j-r2 ) - z-.6z  • 

To  find  the  fluent  of  - —— — ■—  or  (a  — xn)2  x2n-’i. 
in— 1 x 

x2 

38.  When  there  are  several  Terms,  involving'Txuo  or  more  Va- 
riable Quantities,  having  the  Fluxion  of  each  Multiplied  by 
the  other  Quantity  or  Quantities  : 


Take  the  fluent  of  each  term,  a?  if  there  were  only  one 
variable  quantity  in  it,  namely,  that  whose  fluxion  is  contain- 
ed in  it,  supposing  all  the  others  to  be  constant  in  that  term  ; 
then  if  the  fluents  of  all  the  terms,  so  found,  be  the  very  same 
quantity  in  all  of  them,  that  quantity  will  be  the  fluent  of  the 
whole.  Which  is  the  reverse  of  the  5th  rule  for  finding  flux- 
ions : Thus,  if  the  given  fluxion  be  xy  + xj  , then  the  fluent 
of  xy  is  xy,  supposing  y constant  : and  the  fluent  of  Xv  is  also 
xy,  supposing  x constant : therefore  xy  is  the  required  fluent  of 
the  given  fluxion  xy  + Xy  . 

In  like  manner, 


The  fluent  of  ‘xyz-\-xyz  + xy'z  is  xyz. 
The  fluent  of  2xy'x  -f-  x2j'  is  x2y. 

The  fluent  of  \x~2  .xy2  +2x  * fy  is 

The  fluent  of 


x xy  • 

or — is 

y y- 


The  fluent  of 


2axxy  2 — 2y  2axx  ax*g 

y s/y  2y  J y 


39.  JVhetr 


FINDING  OF  FLUENTS, 


323 


39.  When  the  given  Fluxional  Expression  is  in  this  Form — — 

namely,  a Fraction,  including  Two  Quantities,  being  the  Flux- 
ion of  the  former  of  them  drawn  into  the  latter,  minus  the 
Fluxion  vj  the  latter  drawn  into  the  former,  and  divided  by 
the  Square  of  the  latter  : 


Then,  the  fluent  is  the  fraction  -,  or  the  former  quantity  di- 
vided by  the  latter.  That  is,  ’ 

The  fluent  of— — is  -.  And,  in  like  manner, 

y s y 

rp,  a ' n ’Zx'xyi—Oxiyy  . X2 

The  fluent  of — JS  — 

y*  y * ’ 


Though,  indeed,  the  examples  of  this  case  may  be  perform 
ed  by  the  foregoing  one.  Thus,  the  given  fluxion 

^ — ~ — reduces  to  - — — or  - — xj  y~2  ; of  Which, 
yi  _ y y*  y y * ' 

the  fluent  of  - is  - supposing  y constant  ; and 

the  fluent  of  — xjy- 2 is  also  xy~lov  when  x is  constant 

y 

therefore,  by  that  case,  - is  the  fluent  of  the  whole 


. 

40.  When  the  Fluxion  of  a Quantity  is  Divided  by  the  Quanh - 
tity  itself: 

Then  the  fluent  is  equal  to  the  hyperbolic  logarithm  of  that 
quantity ; or,  which  is  the  same  thing,  the  fluent  is  equal  to 
2-30258509  multiplied  by  the  common  logarithm  of  the  same 
quantity. 

So,  the  fluent  of  - or  x~lx,  is  the  hyp.  log.  of  a;. 
x 

The  fluent  of  ?£.  is  2 X hyp.  log.  of  a;,  or  = hyp.  log.  x- . 
x 

The  fluent  of  — , is  a X hyp.  log.  x,  or  = hyp.  log.  of  x2 , 

The  fluent  of  — — , is 
a+.r 

The  fluent  of  is 


41.  Manv 


324 


FINDING  OF  FLUENTS. 


41.  Many  fluents  may  be  found  by  the  Direct  Method  thus 
Take  the  fluxion  again  of  the  given  fluxion,  or  the  second 

X z 

fluxion  of  the  fluent  sought ; into  which  substitute  — for  x 
— for  y , &c.  ; that  is,  make  x,  x,  x,  as  also  y,  y,  y,  kc.  to 

y . .... 

be  in  continual  proportion,  or  so  that  x : x : ■ x : x,  and  - 
y : y : : y : y,  &c.  ; then  divide  the  square  of  the  given  flux- 
ional  expression  by  the  second  fluxion,  just  found,  and  the  quo- 
tient will  be  the  fluent  required  in  many  cases. 

Or  the  same  rule  may  be  otherwise  delivered  thus  : 

In  the  given  fluxion  f,  write  x for  x,  y for  y,  &c.,  and  call 
the  result  g,  taking  also  the  fluxion  of  this  quantity  g ; then 
make  o : f : : g : f ; so  shall  the  fourth  proportional  f be  the 
fluent  sought  in  many  cases. 

It  may  be  proved  if  this  be  the  true  fluent,  by  taking  the 
flu  ion,  of  it  again,  which,  if  it  agree  with  the  proposed 
fluxion,  will  show  that  the  fluent  is  right ; otherwise,  it  is 
wrong. 


EXAMPLE. 

Exam.  1.  Let  it  be  required  to  find  the  fluent  of  nxn_1  x. 

Here  r — nxn~’x.  Write  x , for  x,  then  nxD~1x  or  nxn=G  ; 
the  fluxion  of  this  is  <~  =n2xn~lx  ; therefore  c : f : : g : f, 
becomes  n2xn~1x‘-  nxu~'x  ■ • wxn  : xn  = f,  the  fluent  sought. 

Exam.  2.  To  find  the  fluent  of  xy  + *'y- 

Here  f = xy+xy  ; then,  writing  x for  x and  y for  y , it  is 
xy  + xy  or  2xy  = g ; hence  g = ~xy  + 2x  y ; then  g : 
F : : g : f,  becomes  2 xy  + 2xy  : xy  + x'y  : : 2 xy  : xy  = p. 
the  fluent  sought. 

42.  To  find  Fluents  by  means  of  a Table  of  Forms  of  Fluxions 
and  Fluents. 

In  the  following  Table  are  contained  the  most  usual  forms 
of  fluxions  that  occur  in  the  practical  solution  of  problems, 
with  their  corresponding  fluents  set  opposite  to  them  ; by 
means  of  which,  namely,  by  comparing  any  proposed  fluxion 
with  the  corresponding  form  in  the  table,  the  fluent  of  it  will 
be  found. 


Forms. 


FORMS  OF  FLUXIONS  AND  FLUENTS. 


32  b 


Inns. 

Fluxions. 

Fluents. 

I 

XD~lX 

x*  1 _ 

— or  -xn ■ 
n n 

11 

(a  dz  x°  ')m~ixn~1x 

± — (a  i -3;n)m 

mn 

-- 

: 

— 

— 

V 

11 

-j.mil— lx 

(a  ■+;  X“)m  + 1 

i v 

mna  (a±xn)m 

V 

[a  dz  a;n  )m— 'x 

a-mn-H 

— 1 (a±xn)m 

mna  x™1 

V 

rI 

— 

1 

— 

1 

k 

(myx+nxy  ) X xm-y-‘, 
or  ( — - + — xmi/n 

xmyn 

ma;ra-|.ry^r  f-nx!nJ/n_1i/2r+ rxmi/n21'-1i, 
or  (mxyz-f-nxyz~rrxy'=  )xm-]ya~'  z*-\ 

or  ( — - + — + )xmy°z\ 

K x y z 

xmyDzr 

x _r 

- or  x \r 

JT 

log.  of  x. 

A’t'-'a- 

H+In 

dz  — log.  of  a dz  xn 
n ° 

X — i 
adL:xn 

1 Xn 

— log.  of  — 

na  a±xn 

k 

a — x“ 

1 l0S.of'/“+^1-" 

n^/a  y/  a — y x* 

I 

a + x" 

2 x 

X arc  to  tan  y/  or 

ny/a  a 

1 a — xn 

, X arc  to  cosine  , 
n y/ a a+xn 

X-'n-l.r 
y/  ±a+xn 

~ log.  o{y/xu+y/±a-FxD\ 

Forms ■ 


326  FORMS  OF  FLUXIONS  AND  FLUENTS. 


Forms . 

Fluxions 

fluents. 

XIII 

or^n_Ii 
s/  a-xn 

2 . x11 

- X arc  to  sin.  N — , or 

n w a 

1 2xn 

- X arc  to  vers.  — 

n a 

XIV 

x-\'i 

■ 1 log.  of±x/a±  *“+v/« 

y/  CL  UCn 

n^a  <y  a ± xn  -f-  y/  a 

XV 

x~'x 

v/  — a-\-xu 

2 . . . x" 

X arc  to  secant  ./  — , or 

n v'  a v a 

t . 2a— xn 

X arc  to  cosin. 

n a x a 

XVI 

x^/  dx  — x- 

- circ.  seg.  to  diam.  d & vers,  x 

XVI 1 

cnx 

n log.  c 

XVIII 

xy*  log-  y-r-xy'-y 

y * 

Note.  The  logarithms,  in  the  above  forms,  are  the  hyper- 
bolic ones,  which  are  found  by  multiplying  the  common  loga- 
rithms by  2-302535092994.  And  the  arcs,  whose  sine,  or 
tangent,  &c.  are  mentioned,  have  the  radius  1,  and  are  those 
in  the  common  tables  of  sines,  tangents  and  secants.  Also, 
the  numbers  in,  n,  &c.  and  to  be  some  real  quantities,  as  the 
forms  fail- when  in  = 0,  or  n = 0,  &c. 

The  Use  of  the  foregoing  Table  of  Forms  of  Fluxions  and  Fluents 

43.  In  using  the  foregoing  table,  it  is  to  be  observed,  that 
the  first  column  serves  only  to  show  the  number  of  the  forms  ; 
in  the  second  column  are  the  several  forms  of  fluxions,  which 
are  of  different  kinds  or  classes  ; and  in  the  third  or  last 
column,  are  the  corresponding  fluents. 

The  method  of  using  the  table,  is  this.  Having  aD\ 
fluxion  given,  to  find  its  fluent : First,  Compare  the  given 
fluxion  with  the  several  forms  of  fluxions  in  the  second  co- 
lumn of  the  table,  till  oue  of  the  forms  be  found  that  agrees 
with  it  ; which  is  done  by  comparing  the  terms  of  the  given 
fluxion  with  the  like  parts  of  the  tabular  fluxion,  namely, 
the  radical  quantity  of  the  one,  with  that  of  the  other : and  ; 


FINDING  OF  FLUENTS. 


327 


the  exponents  of  the  variable  quantities  of  each,  both  within 
and  without  the  vinculum  ; all  which,  being  found  to  agree 
or  correspond,  will  give  the  particular  values  of  the  general 
quantities  in  the  tabular  form  : then  substitute  these  particu- 
lar values  in  the  general  or  tabular  form  of  the  fluent,  and  the 
result  will  be  the  particular  fluent  of  the  given  fluxion:  af- 
:er  it  is  multiplied  by  any  coefficient  the  proposed  fluxion  may 


aave. 


EXAMPLES. 


Exam.  1.  To  find  the  fluent  of  the  fluxion  3x3.r. 

This  is  found  to  agree  with  the  first  form.  And,  by  com- 
aring the  fluxions,  it  appears  that  x = x,  and  n — 1 = A, 
vn  ~ t 5 which  being  substituted  in  the  tabular  fluent,  or 

xn , gives,  after  multiplying  by  3,  the  coefficient,  3 X §x3, 
8. 

rfx3,  for  the  fluent  sought. 


xam.  2.  To  find  the  fluent  of  5x2 iy/c2  — x3,  or5 x2x(c3 — x3)2. 

This  fluxion,  it  appears,  belongs  to  the  2d  tabular  form  : 
ir  a = c3,  and — xn=  — x3,  and  n = 3 under  the  vinculum, 
so  m - 1 = i,  or  m=|,  and  the  exponent  11-1  of xn_I  with- 
it  the  vinculum,  by  using  3 for  n,  is  n—  1 = 2,  which  agrees 
iith  x2  in  the  given  fluxion  : so  that  all  the  parts  of  the  form 
■e  found  to  correspond.  Then,  substituting  these  values 
to  the  general  fluent,  - ^ (a  — xn)m. 

becomes  - f X | (c3  - x3)%  = — (c3  — x3)^. 

Exam.  3.  To  find  the  fluent  of  ~—~- 

This  is  found  to  agree  with  the  8th  form  ; where  - - - 

: x“  = -+-  x3  in  the  denominator,  or  n = 3 ; and  the  nume- 
: tor  xn_1  then  becomes  x2,  which  agrees  with  the  numerator 
the  given  fluxion  ; also  a — 1.  Hence  then,  by  substi- 
ting  in  the  general  or  tabular  fluent,  ^ log.  of  a -f-  xn,  it  be- 
mes  i log.  1 -j-  x3. 

Exam.  4.  To  find  the  fluent  of  ax*x- 
Exam.  5.  To  find  the  fluent  of  2 (lO-h*2)2  %x- 
Exam.  6.  To  find  the  fluent  of 


Exam.  7.  To  find  the  fluent  of 


^ 3 

(c2-f.X2)2’ 

3x2  x 
(a  — x)i 


Exam.  8. 


328 


FINDING  OF  FLUENTS. 


Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 

Exam. 


8.  To  find  the  fluent  of  — it. 

Xs 

9.  To  find  the  fluent  of  ^ — x- 

10.  To  find  the  fluent  of  (~^~~)  x3y2 

11.  To  find  the  fluent  of  (- -Hr-)  xy 

'x  3y  J 

12.  To  find  the  fluent  of  — or  - x—  Lr. 

ax  a 


13.  To  find  the  fluent  of  ■ 


ax 


2-2x 

14.  To  find  the  fluent  of  — or 

2x~x-  2 — x 

15.  To  find  the  fluent  of  - ; or  |'z— 


16.  To  find  the  fluent  of 


■r— 3x3  l — 3x2 

3jCjc 

1 — or4 


3 

nX2  & 

17.  To  find  the  fluent  of  — 


18.  To  find  the  fluent  of 

19.  To  find  the  fluent  of 

20.  To  find  the  fluent  of 


2 — X5’ 
2xx 

3 

ax^x 


2i -xs 
Zx'x 


x/1  -f-x* 


21.  To  find  the  fluent  of  — — 


y/x  2-4* 

22.  To  find  the  fluent  of  — ~= 

\Zl  — x*‘ 

23.  To  find  the  fluent  of  — — - 

x/4—x2' 

24.  To  find  the  fluent  of-2jl~~-  — 

y/l  — x2' 


25.  To  find  the  fluent  of 

26.  To  find  the  fluent  of 


v/axz+x|  • 

2x—x 


s/xz-V 


Exam.  27. 


FINDING  OF  FLUENTS. 


329 


Exam.  27.  To  find  the  fluent  of — — ^ . 

\/x2 — ax? 

Exam.  28.  To  find  the  fluent  of  2j;v/2x  — x2. 

Exam.  29.  To  find  the  fluent  of  ax x- 

Exam.  30.  To  find  thp  fluent  of  3 a^x- 

Exam.  31.  To  find  the  fluent  of  2>z*x  log.  ^ -f-  3 xzx~lz  ■ 

Exam.  32  To  find  the  fluent  of  (1  +^3)  xx- 


Exam  33.  To  find  the  fluent  of  (2  + x*)  x 2x 
Exam.  34.  To  find  the  fluent  of  x2x  y/a2  + x2  ■ 


To  find  Fluents  by  Infinite  Series. 


44.  When  a given  fluxion,  whose  fluent  is  required,  is  so 
complex,  that  it  cannot  be  made  to  agree  with  any  of  the 
forms  in  the  foregoing  table  of  cases,  nor  made  out  from  the 
general  rules  before  given  ; recourse  may  then  be  had  to  the 
method  of  infinite  series  ; which  is  thus  performed  : 

Expand  the  radical  or  fraction,  in  the  given  fluxion,  into 
an  infinite  series  of  simple  terms,  by  the  methods  given  for 
that  purpose  in  books  of  algebra  ; viz.  either  by  division  or 
extraction  of  roots,  or  by  the  binomial  theorem,  &c.;  and  mul- 
tiply every  term  by  the  fluxional  letter,  and  by  such  simple 
variable  factor  as  the  given  fluxional  expression  may  con- 
tain. Then  take  the  fluent  of  each  term  separately,  by  the 
foregoing  rules,  connecting  them  all  together  by  their  proper 
signs  ; and  the  series  will  be  the  fluent  sought,  after  it  is  mul- 
tiplied by  any  constant  factor  or  coefficient  which  may  be  con- 
tained in  the  given  fluxional  expression. 

45.  It  is  to  be  noted  however,  that  the  quantities  must 
be  so  arranged,  as  that  the  series  produced  may  be  a con- 
verging one,  rather  than  diverging  : and  this  is  effected  by 
placing  the  greater  terms  foremost  in  the  giveu  fluxion. 
When  these  are  known  or  constant  quantities,  the  infinite 
series  will  be  an  ascending  one  ; that  is,  the  powers  of  the 
variable  quantity  will  ascend  or  increase  ; but  if  the  variable 
quantity  be  set  foremost,  the  infinite  series  produced  will  be 
a descending  one,  or  the  powers  of  that  quantity  will  de- 
crease always  more  and  more  in  the  succeeding  terms,  or  in- 
crease in  the  denominators  of  them,  which  is  the  same  thing. 

; Vol.  II.  43  For 

K 

i y 


330 


FINDING  OF  FLUENTS. 


1 — 3C 

For  example,  to  find  the  fluent  of x- 

r » 1+X  — X2 

Here,  by  dividing  the  numerator  by  the  denominator,  the 
proposed  fluxion  becomes^  — 2xx-{-3x2x  — 5x3x-\-£,xix — kc; 
then  the  fluents  of  all  the  terms  being  taken,  give 
x ~x2-±-x3  -~%x4-}-fx5  —kc.  for  the  fluent  sought. 

Againr  to  find  the  fluent  of  x y/\ — x2. 

Here,  by  extracting  the  root,  or  expanding  the  radical 
quantity  1 — x2,  the  given  fluxion  becomes  - 

x — ix2x  — }x4x — T\x6x~kc.  Then  the  fluents  of  all 
the  terms,  being  taken,  give  x — |r3  — x5  — 7\7x'  — kc. 
for  the  fluent  sought. 

OTHER  EXAMPLES. 


Exam.  1.  To  find  the  fluent 
and  descending  series. 

Exam.  2.  To  find  the  fluent  of 
Exam.  3.  To  find  the  fluent  of 
Exam.  4.  To  find  the  fluent  of 


of  ~-~x  both  in  an  ascending 


in  both  series. 


a-f.  x 

(a— x)2 

1 — x2  . 

~ — ; x- 

l-j~X—  X2 


bi 

Exam.  5.  Given  2 = -— , to  find  2. 

a2  +x2 

a2  -f  X2  . 

Exam.  6.  Given  2 = x to  find  2. 

CL  -J- 

Exam.  7.  Given  2 = 3x  a + x,  to  find  2. 

Exam.  8.  Given  2 — 2.r  y/  a2  -f-  x2,  to  find  2, 

Exam.  9.  Given  2 = 4x  a2  — x 2 , to  find  2. 

Exam.  10  Given  2 = — — — , to  find  2. 

^/X2  — ,2 

Exam.  11.  Given  2 — -x\/a.3—x3,  to  find  2 . 

Exam.  12.  Given  z — — — , to  find  2. 

v/  ax  — xx 

Exam.  13.  Given  z = 2x  \/x 3 + *4  + xB,  to  find  2. 
Exam.  14.  Given  2 = bx^/ax  — xx,  to  find  2. 


To 


FINDING  OF  FLUENTS. 


331 


To  Correct  the  Fluent  of  any  Given  Fluxion. 

46.  The  fluxion  found  from  a given  fluent,  is  always 
perfect  and  complete  ; but  the  fluent  found  from  a given 
fluxion  is  not  always  so  ; as  it  often  wants  a correction,  to 
make  it  contemporaneous  with  that  required  by  the  problem 
under  consideration,  &c.  : for,  the  fluent  of  any  given  fluxion, 
as  x may  be  either  x , which  is  found  by  the  rule,  or  it  may 
be  x + c,  or  x — c,  that  is  x plus  or  minus  some  constant 
quantity  c ; because  both  x and  x ±c  have  the  same  fluxion  x, 
and  the  finding  of  the  constant  quantity  c,  to  be  added  or 
subtracted  with  the  fluent  as  found  by  the  foregoing  rules,  is 
called  correcting  the  fluent. 

Now  this  correction  is  to  be  determined  from  the  nature 
of  the  problem  in  hand,  by  which  we  come  to  know  the  re- 
lation which  the  fluent  quantities  have  to  each  other  at  some 
certain  point  or  time.  Reduce,  therefore,  the  general  fluent- 
tial  equation,  supposed  to  be  found  by  the  foregoing  rules, 
to  that  point  or  time  ; then  if  the  equation  be  true,  it  is 
correct  ; but  if  not,  it  wants  a correction  ; and  the  quantity 
of  the  correction,  is  the  difference  between  the  two  general 
sides  of  the  equation  when  reduced  to  that  particular  point. 
Hence  the  general  rule  for  the  correction  is  this  : 

| 

Connect  the  constant,  but  indeterminate,  quantity  c,  with 
one  side  of  the  fluential  equation,  as  determined  by  the  fore- 
going rules  ; then,  in  this  equation,  substitute  for  the  variable 
quantities,  such  values  as  they  are  known  to  have  at  any 
particular  state,  place,  or  time  ; and  then,  from  that  particu- 
lar state  of  the  equation,  find  the  value  of  c,  the  constant  quan- 
tity of  the  correction. 


EXAMPLES. 

47.  Exam.  1.  To  find  the  correct  fluent  of  z — ax3x- 

The  general  fluent  is  z = ax 4,  or  z = ax4  -f-  c,  taking  in 
the  correction  c. 

Now,  if  it  be  known  that  z and  x begin  together,  or  that 
z is  = 0,  when  x — 0 ; then  writing  0 for  both  x and  z,  the 
general  equation  becomes  0 = 0 + c,  or  — c ; so  that,  the 
value  of  c being  0,  the  correct  fluents  are  z — ax^. 


But 


332 


FINDING  OF  FLUENTS. 


But  if  z be  = 0,  when  x is  = b,  any  known  quantity  ; then 
substituting  0 for  z,  and  b for  x,  in  the  general  equation,  it  be- 
comes 0 = ab4  + c.  and  hence  we  find  c = — ab 4 ; which  be- 
ing written  tor  c in  the  general  fluential  equation,  it  becomes 
^ = ax4  — ab4,  for  the  correct  fluents. 

Or,  if  it  be  known  that  z is  = some  quantity  d.  when  x 
is  = some  other  quantity  as  b ; then  substituting  d for  2,  and 
b for  x , in  the  general  fluential  equation  z — ax 4 + c,  it 
becomes  d — ab 4 + c ; and  hence  is  deduced  the  value  of 
the  correction,  namely,  c = d — ab*  ; consequently,  writing 
this  value  for  c in  the  general  equation,  it  becomes  - - - 

z = ax 4 — ab 4 + d,  for  the  correct  equation  of  the  fluents  in 
this  case. 

48.  And  hence  arises  another  easy  and  general  way  ot 
correcting  the  fluents,  which  is  this  : In  the  general,  equation 
of  the  fluents  write  the  particular  values  of  the  quantities 
which  they  are  known  to  have  at  any  certain  time  or  posi- 
tion ; then  subtract  the  sides  of  the  resulting  particular  equa- 
tion from  the  corresponding  sides  of  the  general  one,  and 
the  remainders  will  give  the  correct  equation  of  the  fluents 
sought. 

So,  the  general  equation  being  2 = ax 4 ; 

write  d for  2,  and  b for  x,  then  d = ab 4 ; 

hence,  by  subtraction,  - 2 — d = ax 4 —ab4 , 

or  2 — ax4  — ab 4 + d , the  correct  fluents  as  before. 

Exam.  2 To  find  the  correct  fluents  of  z = hxx\  z being 
= 0 when  x is  = a. 


Exam.  3.  To  find  the  correct  fluents  of  z = 3x  a -j-  x : 
2 and.t  being  = 0 at  the  same  time. 

Exam.  4.  To  find  the  correct  fluent  of  z = ; sup- 

a + x 

posing  2 and  x to  begin  to  flow  together,  or  to  be  each 
= 0 at  the  same  time. 

2* 

Exam.  5.  To  find  the  correct  fluents  of  z = — — — ; sup- 

a2 

posing  2 and  x to  begin  together. 


Art.  49. 


[ 333  ] 


OF  FLUXIONS  AND  FLUENTS. 


Art.  49.  In  art  42,  &c.  is  given  a compendious  table  of 
various  forms  of  fluxions  and  fluents,  the  truth  of  which  it  may 
be  proper  here  in  the  first  place  to  prove. 

50.  As  to  most  of  those  forms  indeed,  they  will  be  easily 
proved,  by  only  taking  the  fluxions  of  the  forms  of  fluents, 
in  the  last  column,  by  means  of  the  rules  before  given  in 
art.  30  of  the  direct  method  ; by  which  they  will  be  found  to 
produce  the  corresponding  fluxions  in  the  2d  column  of  the 
table  Thus,  the  1st  and  2d  forms  of  fluents  will  be  proved 
by  the  1st  of  the  said  rules  for  fluxions  ; the  3d  and  4th  forms, 
of  fluents  by  the  4th  rule  for  fluxions  ; the  5th  and  6th  forms, 
by  the  3d  rule  of  fluxions  : the  7th,  8th,  9th,  10,  12th,  14th 
forms,  by  the  6th  rule  of  fluxions  : the  17th  form,  by  the  7th 
rule  of  fluxions  : the  18th  form,  by  the  8th  rule  of  fluxions. 
So  that  there  remains  only  to  prove  the  11th,  13th,  15th,  and 
16th  forms. 

51.  Now,  as  to  the  16th  form,  that  is  proved  by  the  2d  ex- 
ample in  art.  98,  where  it  appears  that  x^/^dx—x2)  is  the 
fluxion  of  the  circular  segment,  whose  diameter  is  d,  and 
versed  sine  x.  And  the  remaining  three  forms,  viz.  the  1 1th, 
13th,  and  15th,  will  be  proved  by  means  of  the  rectifications 
of  circular  arcs,  in  art.  96. 

52.  Thus,  for  the  1 1th  form,  it  appears  by  that  art.  that  the 
fluxion  of  the  circular  arc  z,  whose  radius  is  r and  tangent  t, 

r%  1 XT  JL  n 

is  z = — . — . Now  put  t — x2n,  or  f2  = x , and  a = r2  : 

r2-\-t  r 

. . JLn— i n . ,-2  i 

then  is  t ~ \nx2  x,  and  r2  t2  = a + x , and  z = — — - 

F-  -f*  t- 

-I71-1 . 1 . 

= — ; hence  — — = — = - z , and  the  fluent  is 

a + ’ a+x"  ian  an 

2 z 2 . -L72  2 

— = — X arc  to  radius  a and  tang,  x2  or  = X arc. 

an  na  v n<ya 

xn 

to  radius  1 and  tang.  < / — , which  is  the  first  form  of  the 
v a 

fluent  in  n°.  xi. 

53.  And,  for  the  latter  form  of  the  fluent  in  the  same  n°  ; 

because  the  coefficient  of  the  former  of  these,  viz.  —7 — is 

n^/a 

double  of the  coefficient  of  the  latter,  therefore  the  arc 

n-Sa 

jn  the  latter  case,  must  be  double  the  arc  in  the  former. 
But  the  cosine  of  double  an  arc,  to  radius  1 and  tangent  ?,  is 

1—t  2 
1+*®  ; 


334 


FLUXIONS  AND  FLUENTS. 


l— t* 


l+tz 


; and  because  t2 


xn 


by  the  former  case,  this  substi- 


tuted for  t 2 in  the  cosine 


1— *2 


it  becomes  C ■,  X , the  co- 


l-J-;2  ii  n 5 

sine  as  in  the  latter  case  of  the  11th  form. 

54.  Again,  for  the  first  case  of  the  fluent  m the  13th 
form.  By  art.  61,  the  fluxion  of  the  circular  arc  z,  to  radius  r 


and  sine  y,  is  z 


ry 


or  = 


v'U2— y2)  ✓(!— y2) 

Xn  Xn 

Now  put  y — s/  — , or  y2  = — ; hence  ( I 


to  the  radius  1. 


V2)  = 


a/(1~T‘):=v/^xv'  («—*")>  and  y = ^/-XUz' 


then  these  two  being  substituted  in  the  value  of  z , give  ~ 

or  ^(i— yT)~=|  X^r^)  ; consequently  the  given  fluxion  I 
J|»-i . 


2 2 

ris  =-  z,  and  therefore  its  fluent  is  — z,  that  is 
v'C  a—xn)  n n 


— X arc  to  sine  — , as  in  the  table  of  forms,  for  the  first 

n a 


case  of  form  xm. 


55.  And,  as  the  coefficient  — , in  the  latter  case  of  the  said 


form,  is  the  half  of  — the  coefficient  in  the  former  case,  i 

n 

therefore  the  arc  in  the  latter  case  must  be  double  of  the  arc 
in  the  former.  But,  by  trigonometry,  the  versed  sine  of 
double  an  arc,  to  sine  y and  radius  1,  is  2y2  ; and,  by  the 


2xn  1 

former  case  2 y2  ~~ — ; therefore  — X arc  to  the  versed  sine 


- — is  the  fluent,  as  in  the  2d  case  of  form  xm. 
a 

56.  Again,  for  the  first  case  of  fluent  in  the  15th  form. 
By  art  61,  the  fluxion  of  the  circular  arc  z,  to  radius  r 

> or  = — rr^ — — to  radius  1. 


and  secant  s,  is  z — 

sv'U2  ■ 
4" 

xn 

■v  ~ 


Now,  put  s 

Jn 


■ r*y 
or  s2 


■fv'U2— 1) 


Sa 


— ; hence  s^/fs5-  1)= 

a 

-L 


J"  . 1 ir‘~l . 

- — y/  (- 1)  = — */(xn—a),  and  « =»/—  X \nx  x ; 

a a ' a a 

then  these  two  being  substituted  in  the  value  of  z,  give  -or 
— "%/a  \ x x . C0Dsequent]y  the  given  fluxion 


v/(xn-a) 


V(i2“  1)  2 

x~~^  'x  2 , ^ 4 

— t t ■ — -■ — • z > and  theref.  its  fluent  is  — z,  that  is  -::— 

ny/a  iiy/a  n^a 

X arc 


FLUXIONS  AND  FLUENTS.  335 

#' 
ccn 

X arc  to  secant  */  — , as  in  the  table  of  forms,  for  the  lirst 

a 

case  of  form  xv. 

57.  And,  as  the  coefficient  — — , in  the  latter  case  of  the 

n*ya 

2 

said  form,  is  the  half  of the  coefficient  of  the  former 

n-J  a 

case,  therefore  the  arc  in  the  latter  case  must  be  double  the 
arc  in  the  former.  But,  by  trigonometry,  the  cosine  of  the 

2 

double  arc  to  secant  sand  radius  1,  is 1 ; and,  by  the 

s3 

former  case,  — — 1=  — — 1 = ~ — — ; therefore  —1-  X 

s2  x n xn  n</a 

arc  to  cosine  is  the  fluent,  as  in  the  2d  case  of  form  xv. 

xn 

<2  . a 

Or,  the  same  fluent  will  be  — — X arc  to  cosine  */—  ,be- 
’ n^/a  v xn 

cause  the  cosine  of  an  arc,  is  the  reciprocal  of  its  secant. 

58.  It  has  been  just  above  remarked,  that  several  of  the 
tabular  forms  of  fluents  are  easily  shown  to  be  true,  by  taking 
the  fluxions  of  those  forms,  and  finding  they  come  out  the 
same  as  the  given  fluxions.  But  they  may  also  be  deter- 
mined in  a more  direct  manner,  by  the  transformation  of  the 
given  fluxions  to  another  form.  Thus,  omitting  the  first 
form,  as  too  evident  to  need  any  explanation,  the  2d  form  is 
z = (a  + jrn)m-1.rn-br,  where  the  exponent  ( n — 1)  of  the 
unknown  quantity  without  the  vinculum,  is  1 less  than  ( n ), 
that  under  the  same.  Here,  putting  y ==  the  compound 

• • • • • 

quantity  a x”  : then  is  y — nxn~lx,  and  z = ; hence 

vm  rn  jl  v-»'\ m 

by  art.  36,  z,  — i i-as  in  the  table. 

inn  mn 

59.  By  the  above  example  it  appears  that  such  form  of 
fluxion  admits  of  a fluent  in  finite  terms,  when  the  index 
(n—  1)  of  the  variable  quantity  (x)  without  the  vinculum, 
is  less  by  1 than  n,  the  index  of  the  same  quantity  under  the 
vinculum.  But  it  will  also  be  found,  by  a like  process,  that 
the  same  thing  takes  place  in  such  forms  as  (a  + xn)mxcn~1  x, 
where  the  exponent  ( cn  — 1)  without  the  vinculum,  is  1 less 
than  any  multiple  (c)  of  that  ( n ) under  the  vinculum.  And 
further,  that  the  fluent,  in  each  case,  will  consist  of  as  many 
terms  as  are  denoted  by  the  integer  number  c ; viz.  of  one 
term  when  c = 1,  of  two  terms  when  c = 2,  of  three  terms 
when  c = 3,  and  so  on." 

60.  Thus,  in  the  general  form,  z — (a  + xn)mxcn-lx, 
putting  as  before,  a -j-  xn  = y ; then  is  xn  — y — a,  and  its 


fluxion 


336 


FLUXIONS  AND  FLUENTS. 


fluxion  rixn~xx  = y,  or  xn  lx  — and  x01-^  or  xcn~n 

J n 

xn~xx  — — ( y — a)c~ 1 y ; also  (a  -J-  xn')m  = ym  : these  val- 
ues being  now  substituted  iu  the  general  form  proposed, 
give  z = - {y  — a)c~xymy.  Now,  if  the  compound  quantity 

( y — a)c — 1 be  expanded  by  the  binomial  theorem,  and  each 
term  multiplied  by  ymy,  that  fluxion  becomes 

1,  ..  c—l  . . c— l c-  2 


= d'f 


y — 


aym*c~2y 


+ _ a-ym^~ry  — 


n “ 1 y 1 1 2 

&c.) ; then  the  fluent  of  every  term,  being  taken  by  art.  36,  it  is 

1 ,ym*c  c-  1 aym'c-i  c-1  c- 2 a2ym*J- 2 

n y/i-J-c  1 m-f-c—  1 1 ° ***  ° 


V-,/ 


+ 


c-l.c  — 2 


2 m + c—  2 
c — l c—  2 .c—  ; 


a 3 

2.3^3 


c — l 

</— 1 y 1 d — 2 2_y2  — t 

&c.),  putting  d = m + c,  for  the  general  form  of  the  fluent ; 
where,  c being  a whole  number,  the  multipliers  c — 1,  c— 2, 
c-3,  &c.  will  become  equal  to  nothing,  after  the  first  c terms, 
and  therefore  the  series  will  then  terminate,  and  exhibit  the 
fluent  in  that  number  of  terms  ; viz.  there  will  be  only  the 
first  term  when  c = 1,  but  the  first  two  terms  when  c = 2, 
and  the  first  three  terms  when  c = 3,  and  so  on — Except 
however  the  cases  in  which  m is  some  negative  number  equal 
to  or  less  than  c ; in  which  case  the  divisors,  m+c,  m + c — 1, 
m + c—  2,  &c.  becoming  equal  to  nothing,  before  the  multi- 
pliers c—l,  c — 2.  &c.  the  corresponding  terms  of  the  series, 
being  divided  by  0,  will  be  infinite  : and  then  the  fluent  is 
said  to  fail,  as  in  such  case  nothing  can  be  determined  from  it.  „ 


61.  Besides  this  form  of  the  fluent,  there  are  other  methods 
of  proceeding,  by  which  other  forms  of  fluents  are  derived, 
of  the  given  fluxion  z — («  + xn)mxcn~lx,  which  are  of  use 
when  the  foregoing  form  fails,  or  runs  into  an  infinite  series  ; 
some  results  of  which  are  given  both  by  Mr.  Simpson  and  Mr. 
Landen.  The  two  following  processes  are  after  the  manner 
of  the  former  author. 

62.  The  given  fluxion  being  ( a + xn)mxen~1x  ; its  fluent 

may  be  assumed  equal  to  (a  + multiplied  by  a general 

series,  in  terms  of  the  powers  of  x combined  with  assumed 
unknown  coefficients,  which  serites  may  be  either  ascending 
or  descending,  that  is,  having  the  indices  either  increasing  or 
decreasing  : 

viz.  (a+xn)m+1  X (ax’"  + Bxr-S  + cxT~ u + r>xr~3j  + &c  ), 
or  (a+x»)m+1  X (ax'-  4-  bxt*s  + cxrtJJ  + D.Xr+3,  + &c.). 

And 


FLUXIONS  \ND  FLUENTS. 


33*7  ' 


And  first,  for  the  former  of  these,  take  its  fluxion  in  the 
usual  way,  which  put  equal  to  the  given  fluxion  (a  -f-  xn)m 
xcn— Ir,  then  divide  the  whole  equation  by  the  factors  that 
may  be  common  to  all  the  terms  ; after  which,  by  comparing' 
the  like  indices  and  the  coefficients  of  the  like  terms,  the 
values  of  the  assumed  indices  aud  coefficients  will  be  deter- 
mined, and  consequently  the  whole  fluent.  Thus,  the  former 
assumed  series  in  fluxions  is, 

»(m- (-  l)xn~‘*  (o  + *n)m  X (-ix7-  -1 - + cxr~is  &c.)  + 

( a-\-xn)m^lx  X (rAXr_l+  (r— s)  exr~s~  -f-  (r  --  2$)cxT~’,-1—L 
&c.)  ; this  being  put  equal  to  the  given  fluxion  (a-j-xB)!na:cn_1,r, 
and  the  whole  equation  divided  bv  {a-\~  xn)mx~lx  there  results 
w(m-f 1 )xn  X (ax7*  + bxt~j  + ex'--1"  + dx7""35  + &c.)  > _ cn 
+ (a-{-xn)  X (tax'  + (r — s)Bxr-s-j-(r  — 2s)cxr_2i  &c.)  $ X 
Hence,  by  actually  multiplying,  and  collecting  the  coefficients 
of  the  like  powers  of  x,  there  results 
z(m- f-l)  ) AXr+n-\~n(Tti-\-  1 ) ( Bxr*n~s±n(m*lJ  ? cxr+n~^&c. 

-j-r  S 4 rs  ) + r — 2s  \ 

—xcn  ..  + ...  .raxxr  . , . -f-(r  — s)aBx7*“ f &c. 

Here,  by  comparing  the  greatest  indices  of  x,  in  the  first  and 
second  terms,  it  gives  r -f-  n = cn,  and  r -f-  n — s = r ■ 
which  give  r = (c  - 1 )n,  and  n = s.  Then  these  values 
being  substituted  in  the  last  series,  it  becomes 
(e  -{-/«)  nAXc0  4~  (c-j-  m — l)raBx<'n“n-t-(c-f-w»~ 2)ncxcn~ 2n&c. 

xC71+(c—  V)naxxcn~n-\-(c ~‘2')nn.BXcn~ln  &c. 

Now,  comparing  the  coefficients  of  the  like  terms,  and  put- 
ting c -f-  m = d,  there  results  these  equalities  : 

1 c—l-aA  c-l-a  c — 2’OB  ,c — l.c  — 2.q* 


=0. 


0. 


" dn ;B' 


d-  1 ^ d—l.dn’  d- 2 1 d-\.d—<2-dn> 

&c.  ; which  values  of  a,  b,  c,  &e.  with  those  of  r and  s,  being 
now  substituted  in  the  first  assumed  fluent,  it  becomes 
^a+xn)rn+\xcn—n  ^ c-l.Oj  C — 1-C — 2m2  c—  l.C  — 2.C— 3.a3 

dn  ^ 1 ” d-  l.xn-  d- \.cl  — 2-x2n~~ d-l.cl- 2-d-j.x 3 n 

4-  &c.  the  true  fluent  of  (a  -j-  xn)mxcn~ljd,  exactly  agreeing 
with  the  first  value  of  the  19th  form  in  the  table  of  fluents 
in  my  Dictionary.  Which  fluent  therefore,  when  c is  a whole 
positive  number,  will  always  terminate  in  that  number  of 
:erms  ; subject  to  the  same  exception  as  in  the  former  case. 
Thus,  if  c — 2,  or  the  given  fluxion  be  (a  -f-  xn)'nxZn~lx ; 
hen,  c + m or  d being  = m - f-  2 the  fluent  becomes 


=4* 


(a-f-xn)m+lxn  . 


ax~»,  (a-fx’n)7n+I  (?n-f  l)m  — a 


(1-  -—)  = 
v m4-  1' 


Xr 


(w+2lra  -v-  n '"m  + !• 

knd  if  c = 3,  or  the  given  fluxion  be  (a  -f-  xn)mx3n~lx  -T 
hen  tn  -f  c or  d being  = rn  + 3,  the  fluent  becomes 

a-\-xn)m^ix2n  ^ 2ax~»  , 2a2 x~ 2n  N (a-tx")"1*1^  f Xm 

* 1 ~ 2 ''■m-i-  J ~ ^ V 


(m-p3)rc 
Yob.  II. 


44 


v m+3 
2e.rr‘ 


338 


FLUXIONS  ANB  FLUENTS. 


2axn 


2a2 


-).  And  so  on,  when  c is  other 


m+  3.772 -f-2  ' m + om +2.m-bl 
whole  numbers  : but,  when  c denotes  either  a fraction  or  a 
negative  number,  the  series  will  then  be  an  infinite  one,  as* 
none  of  the  multipliers  c — 1,  c — 2,  c — 3,  can  then  be  equal 
to  nothing. 


63.  Again,  for  the  latter  or  ascending  form,  (a  + ;rn)m+1  X 
(axt  + Bxr*s  + cxr*2t  + Dxr+3*  4-  kc.),  by  making  its  fluxion 
equal  to  the  proposed  one,  and  dividing,  k c.  as  before,  equa- 
ting the  two  least  indices,  kc.  the  fluent  will  be  obtained  in  a 
different  form,  which  \yill  be  useful  in  many  cases,  when  the 
foregoing  one  fails,  or  runs  into  an  infinite  series.  Thus,  if  jj 
r + s,  r + 2s,  kc.  be  written  instead  of  r — s,  r — 2s,  I 
kc.  respectively,  in  the  general  equation  in  the  last  case, 
and  taking  the  first  term  of  the  2d  line  into  the  first  line, 
there  results 


r4n+w(ni+1) 

* +r+s 


— xcn+«(m+l 
+»■ 

+raAzr+(r+s)aB:cr-‘i4-(r+2s)acxr+2i  kc 


'+S  &Ct  \ = 
£C.  ) 


0. 


Here,  comparing  the  two  least  pairs  of  exponents,  and  the 

coefficients,  w'e  have  r — cn,  and  s = n ; then  a = - =—  ; 

ra  cna 

r + «(m+l)  c+7n-fl  a c+ot+1 


,B  = — 


a(r+«) 


c+1 


c+tb+2  _ c+m+l.c+m+  2 
(c+2)a  c.c+1  • c+2.na3 


kc. 


i (c  + l)cnas  ’ 

Therefore,  denoting, 
of  the  same  fluxion 


c + m by  d,  as  before,  the  fluent 
(a+x")"^”1- \r,  will  also  be  truly  expressed  by 
(a+ xn)m*xxcn  (\  d + l.jn  .tf-fl.dg-2.x2" 


(l  c. 


- + 


V'fa  + x) 


(a  + x)2 

This 


example  being  compared  with  the  general  form 
xcn-ij.  (a  + x»)m,  in  the  several  corresponding  parts  of  the 
first  series,  gives  these  following  equalities  : viz.  a = a,  n— 1, 
cn  — 1 = 1,  ore  — 1 = 1 , or  c =;  2 ; m = — 4 ; y = a + x, 

d — m 


-kc.)-, 


cna  'l  c+1.  a 1 e+  1 • c + 2 . a2 

agreeing  with  the  2d  value  of  the  fluent  of  the  19th  form  in 
my  Dictionary.  Which  series  will  terminate  when  d or  c+m 
is  a negative  integer  ; except  when  c is  also  a negative  in- 
teger less  than  d ; for  then  the  fluent  fails,  or  will  be  infinite, 
the  divisor  in  that  case  first  becoming  equal  to  nothing. 

To  show  now  the  use  of  the  foregoing  series,  in  some  ex- 
amples of  finding  fluents,  take  first, 

64.  Example  1.  To  find  the  fluent  of  — - or  6xjr 


FLUXIONS  AND  FLUENTS. 


339 


^ny 


3 ' 1 o 

«(.+«)*,  I =4 


c - 1 


d — m-src=  2 -i=f, 

- = ■ 2a  ■ ; here  the  series  ends,  as  all  the  terms  after  this 
y a +x 

become  equal  to  nothing,  because  the  following  terms  con- 
tain the  factor  c — 2 = 0.  These  values  then  being  substi- 


,1  c—  1 


tuted  in  — it  becomes  (a  -f-  x)2  X 

n x d d— 1 y ' s ' 

.2  2a  . .2 a + 2x  . , , .i  2x—4a  . , . 

(— 2“>  X (“  + *>S  = 3 + *> 

which  multiplied  by  6,  the  given  coefficient  in  the  proposed 
example,  there  results  (4x  — 8a)  . (a  + x),  for  the  fluent 
yequired. 


17. 


Exam.  2.  To  find  the  fluent  of 
3i,/(a2 +**)  i 


XT9 


(a2  + x2)2  X 3x~6x. 


The  several  parts  of  this  quantity  being  compared  with  the 
corresponding  ones  of  the  general  form,  give  a — a2 ,n  — 2, 

in  = i,  cn  — 1 or  2c  — 1 = — 6,  whence  c = — — = — # 
" 2 

and  d = m-\-c=-\  — f = — f = — 2,  which  being  a nega- 
tive integer,  the  fluent  will  be  obtained  by  the  3d  or  last  form 
ef  series  ; which  on  substituting  these  values  of  the  letters, 

Ua*+x*y*x-s„  rl  -!•**  3(aa  +x*)\  2x2 

s,ves  — Ts? — X(.i~z^r)= 


3a2 


-a2  ^~X~5  ^or  re<lu^re(l  fluent  of  the  proposed 

fluxion. 


66.  Exam.  3.  Let  the  fluxion  proposed  be 

5x3n—lL  I 

~ yr,  n{  = 3{b  + xn)-2  x3n~lx. 

v/CA-f-xn) 

Here,  by  proceeding  as  before,  we  have  a = b,  n = n, 
m = — 4,  c — 3,  and  d = c + m = f ; where  c being  a 
positive  integer,  this  case  belongs  to  "the  2d  series  ; into 
which  therefore  the  above  values  being  substituted,  it  becomes 

5(6-|-xn)2x2n^  ,1  2 b , 2.1.62  „ , „N  w3x*n“ 4bxn+862 

in  1 ixn 


+: 


3 l rr'ln 
2*2X 


■— 2v/(6+xT1)  X- 


3a 


67.  Exam.  4.  Let  the  proposed  fluxion  be  5 z2)2^  3z  ■■ 

Here,  proceeding  as  above,  we  have  a — i,  n = 2,  to  = 4> 
ca  — 1 or  2c  — 1 = — 8,  and  c — — i,  x—  — z,d—  c -f- 
m = — 3,  which  being  a negative  integer,  the  case  belongs 
"to  the  3d  or  last  series  ; which  therefore,  by  substituting 

these 


340 


FLUXIONS  AND  FLUENTS. 


these  values,  becomes— 3-~  2 X — — 4 — - ' - — . 

- 7 . iZ7  - 'l  —5.1  — i.  — 3 . JL 


35( „ , 12z*  , 24z4  ^ -3(i 


22— X (5+  12z2  4*24z4 ), 


5 7 7z’ 

the  true  fluent  of  the  proposed  fluxion.  And  thus  may  many 
other  similar  fluents  be  exhibited  in  finite  terms,  as  in  these 
following  examples  for  practice. 

Ex.  5..  To  find  the  fluent  of  — 3x3x^/(a*  — x2). 

Ex.  6.  To  find  the  fluent  of  — 6x5-r  . (a2  — xfe)~2 

Ex.  7.  To  find  the  flu.  of  — x—  or  (a— xn'fx~2n^x 

\ 7.  1 

X2n-1 


68.  The  case  mentioned  in  art.  37,  viz.  of  compound  quan- 
tities under  the  vinculum,  the  fluxion  of  which  is  in  a given 
ratio  to  the  fluxion  without  the  vinculum,  with  only  one  varia- 
ble letter,  will  equally  apply  when  the  compound  quantities 
consist  of  several  variables.  Thus, 

Example  1.  The  given  fluxion  being  (4x^  -f-  8 yy)  X 

(jc - -f-2?/2),  or  (4x’x-\-8yy  ) X (x3+2y2)2,  the  root  being 
x £4-2y7,  the  fluxion  of  which  is  2xx~i~  \y'y  Dividing  the 
former  fluxional  part  by  this  fluxion,  gives  the  quotient  2 : 
next,  the  exponent  \ increased  by  1,  gives  f : lastly,  dividing 

I—" 

by  this  f,  there  then  results  f {x2 +2y2) 2 , for  the  required 
fluent  of  the  proposed  fluxion. 


Exam.  2.  In  like  manner  the  fluent  of 

(t5+  y4  + z6)2  X (6x4  + 12 y3y  -f-  182sz)  is 
(^+y«+z«)34lx(6TJ;  + 12y3y--H825:)  = £ * f. 

(2x^+4y3-  -)-6z««  )Xf  4(1  ^ ^ ) 

Exam.  3.  In  like  manner,  the  fluent  of 

2x-  (iy-  -f-  xy'y  + xsx)  ^ / ( x 2 + 2 y2),  is  A (x4  + 2 x2y2)2. 
69.  The  fluents  of  fluxions  of  the  forms 

OC^  j jy  Xcn~~l ^ 

— v — , : — &c.  or  &c.  where  c and  n are  whole 

x&a  xs  ±a2  xn  o.n 

numbers,  will  be  found  in  finite  terms,  by  dividing  the  nume- 
rator by  the  denominator,  using  the  variable  letter  x as  the 
first  term  of  the  divisor,  continuing  the  division  till  the  powers 
of  x are  exhausted  ; after  which,  the  last  remainder  will  be 
the  flu-  ion  of  a logarithm,  or  of  a circular  arc,  &c. 

Exam.  I.  To  find  the  fluent  of  or  x*  . 

a-\-x  x+a 


FLUXIONS  AND  FLUENTS. 


341 


Bv  division,  -x  ■ = x — — — , where  the  remainder  — — is 
J x-fa  x-fa  x-fa 

evidently  = a X the  fluxion  of  the  hyperbolic  logarithm  of 
a + x : therefore  the  whole  fluent  of  the  proposed  fluxion 
is  a;  — a X hyp.  log.  of  (a  + x).  in  like  manner  it  will  be 
found  that, 


Ex.  2. 

The 

fluent 

of 

XX 

x — a ’ 

is  x-\-a  X hyp.  log 

0 
*** 

1 

Ex.  3. 

The 

fluent 

of 

Xx 

a—* 

is  — x — a X hyp.  log.  of 

|a  — x). 

• 

Ex.  4. 

The 

fluent 

of 

X2x 
a •+■  x' 

is  ix2  - ax  + x2 

X hyp.  log. 

of  (a  -f-  x 

')• 

Ex.  5. 

The 

fluent 

of 

X2'x 

a—x ’ 

is  — \x2  — ax  - 

- a2  X hyp. 

log  of  (a 

-x). 

X2x 

x—a 

Ex.  6. 

The 

fluent 

of 

is  \x2  ax  a2 

X hyp.  log. 

of  (x — a) 

Ex.  7. 

The 

fluent 

of 

X3x 

is  ix3  — fax2  -fa 

2x  — a3  X 

x +a 

hyp.  log. 

of  (x-f  a). 

/ 

Ex.  8. 

The 

fluent 

of 

X3x 

x—a' 

is  ix3  -f-  fax2  + 

a2x  + a3  X 

i*3 


±ax2  — a2x  -f- 


hyp.  log.  of  (a?  — a). 

Ex.  9.  The  fluent  of——,  is 
a—  x 

a3  X hyp.  log.  of  ( a—x ). 

Ex.  10.  The  fluent  of  is  fx4  — lax3  -f  i a2x2  — 

a3x+a4  X hyp.  log.  (a+x). 

Xn'x  . xn 


Ex.  11.  The  fluent  of 


“ 


3^-3 


ai- 


ls 


ax 71-1  , a2xn~2 

n—  1 n — 2 


n — o 


+ &ic.  rfc  a"  X h.  1.  (a-fx). 


2 3 xn—3 


&c.  — a"  X h.  1.  (a  — x) 

Xnx 


n — 3 
Ex.  13.  The  fluent  of 

i 

1 3a?n— 3 


V 

x’  13 

xn 

axn~l 

a2xn~ 2 

n 

n — 1 

1 

a 

1 

to 

- x). 

xn 

+ 

ax 71—1 

a2xn~ 2 

• a’  18 

n 

72 — 1 

+ ^ ~ 
72 — 2 

n—  3 


&c.  + an  X h.  1.  (x  — a). 


x2  j 


Ex.  14.  The  fluent  of  — „ 

x2i -«3 


= (by  division^  ■ 


a2x 


x2  ia2’ 


is 


342 


FLUXIONS  AND  FLUENTS. 


is,  (by  form  1 1 this  vol.)  x — cir.  arc  of  radius  a and  tang.  * 

In  like 


or  x ±a  X cir.  arc  of  rad.  1 and  cosine  - — — 
manner, 


a*x 


Ex.  15.  The  fluent  of-^  — — or  of  — x + —— - — , is  

a2  -x2  r a2  -x2’ 

x -j-  ia  X h.  1.  by  form  10.  And 

a—x-  J 


Ex.  16.  The  fluent  of  — —x  4-  - — , isx-f^-aX 

*2— a2  x2-a2  ’ 2 


a2x 


OC  CL 

hyp.  log.  — — , by  the  same  form. 


X-f-  a 

70.  In  like  manner  for  the  fluents  of 


. 0 2 ,±x* 


Thus, 


Ex.  17.  The  fluent  of  ■ — = x2x  — a2x  -J- 

o2+x2  ^fl2+xa» 

is  by  form,  ix3  — a2x  -\-a2  X cir.  arc  to  rad.  a and  tang,  x, 

Q 2 — ,x  2 

or  ix3  — a2x-{-la3  Xcir.  arc  to  rad  1 and  cosine — . And 

3 a*-fx  2 


Ex.  18.  The  fluent  of  -£^2,  = ~x2i~a2x  + 


a+x 


- — ix3  — a2x  + ia3  X hyp.  log.  — — , by  form  10.  Also 


Ex.  19.  The  fluent  of  * — = x2x  + a2  x -\ — J , is 

OC ^ > CL  * X*  — - CL 2 


lx3  -f-  + £a3  X hyp.  log.  by  form  10 


71.  And  in  general  for  the  fluent  of 


Xnx 


where  n is 


x2  7k a2 

any  even  positive  number,  by  dividing  till  tbe  powers  of  x 
in  the  numerator  are  exhausted,  the  fluents  will  be  found  as  be- 
fore. And  first  for  the  denominator  x2  + a2,  as  m 

Ex.  20.  For  the  fluent  of  -r^--  = (by  actual  division) 


xn~2x  — a2xn~*x  + a4xn_6  — Sic.  ± an~2x  + 


the 


x2  -J-a* 

number  of  terms  in  the  quotient  being  in,  and  the  remainder 
+ ~ , viz.  — or  + according  as  that  number  of  terms  is 

odd  or  even.  Hence,  as  before,  the  fluent 

. {7  2 jrn — 3 

is  — &lc.  . . . ± an~2x  =p  a"-2  X arc  to  rad. 


n—  1 


#n— 1 a2  Xn~3 

a and  tan  x,  or — (-  Sic.  . . i an_2x  qz  £an_1  X 


n — 1 n—c 


arc  to  rad.  1 and  cos. 


a2  —x2 


a2 


Ex.  21. 


.FLUXIONS  AND  FLUENTS, 


343 


Ex.  21.  In  like  manner,  the  fluent  of 


13 


xn~  i a2x’ 1—3  a^x”— 5 

7Z—  l n—S  n-  5 

xnx 


&c.  ian_1  X hyp.  log. 


a2  —x* 

a-j-x 


Ex.  22.  And  of 


X hyp.  log. 


x 2 — a2 


IS 


xn~l  . a 2 xn — 3 


71—1  71—3 


+ • &c.  + |a»-i. 


* + a 

72.  In  a similar  manner  we  are  to  proceed  for  the  fluents  of 
-,  when  n is  any  odd  number,  by  dividing  by  the  de- 


a2  & x2’ 

nominator  inverted,  till  the  first  power  of  x only  be  found 
in  the  remainder,  and  when  of  course  there  will  be  one  term 
less  in  the  quotient  than  in  the  foregoing  case,  when  n was 
an  even  number  ; but  in  the  present  case  the  log.  fluent  of 
the  remainder  will  be  found  by  the  8th  form  in  the  table  of 
fluents. 

jpfl  • 

Ex.  22.  Thus,  for  the  fluent  of -r- — where  n is  an  odd 

Ex2  ha3 
number,  the  quotient  by  division  as  before,  is  xn~2'x  — a2xn~ 4 
i + a*xn-6x  — &c.  ±.  an~3xi,  the  number  of  terms  being 

m~  , and  the  remainder  q:  ‘x’l+L Therefore  the  fluent  i§ 

n2  Xn—Z  an~~3X2 

' &c ± a-— — =p  X h.  1.  + a2. 


2 

X’i-l 


71—1  71  — 3 

Ex.  23.  The  fluent  of 


is  obtained  in  the  same 


2 

Xnx 
x 2 — a2 

manner,  and  has  the  same  terms,  but  the  signs  are  all  positive, 
and  the  remainder  is  -f-  J-a"-1  X hyp.  log.  x2  — a2. 

Ex.  24.  Also  the  fluent  of  is  still  the  same,  but  the 

a2  — x2 

signs  are  all  negative,  and  the  remainder  is  — ±an-'  X hyp. 
log.  a2  — x2.  Hence  also, 

Ex.  25.  The  fluent  of  is  i-^2  — ^aa  X hyp.  log. 

of x2  -j -a2. 

Ex.  26.  The  fluent  of  - is  %x2  -f-  ±a2  X hyp.  log,, 
of  x2  — a2. 

Ex.  27.  The  fluent  of 
log.  of  a 2 — x2 . 

Ex.  28.  The.f 
hyp.  log.  x2  -f-  a2. 


*3 


a 2 


— is  — \x*  — la2  X hyp. 


Ex.  28.  The. fluent  of  , is  i-r*  — \a2x 2 -fia*  X 


Ex.  2S 


44 


FLUXIONS  ANTD  FLUENTS. 


Ex.  29.  The  fluent  of 


hyp.  log.  x 2 — a 2 
Ex.  30.  The 
hyp.  log.  a-  — x 


X2  — a 2 


> is  + ia2x2  + ia 4 X 


Ex.  30.  The  fluent  of  — , is  — ixi  — \a-xz  — ia*  X 

a2  — X2 


73.  £.r.  31.  In  a similar  manner  may  be  found  the 


fluents  of  where  c is  any  whole  positive  number,  by- 


dividing  till  the  remainder  be  — which  can  always 


be  done,  and  the  fluent  of  that  remainder  will  be  had  by  the 
8th  form  in  this  vol.  Thus,  by  dividing  first  by  jn  + a",  the 
terms  are,  xcn~n~'x  — anxcn~2r‘~1x  + ainxcn~ 3n~1x  — -f- 
&c.  till  the  last  term  be  a^-,^nx^-d>-'1,  and  the  remainder 

Qdnx(c~  d)n—  1^  q(c— l)«;rn~  1 


when  d is  = c — 1 , or  1 less  than 


ar»  a" 

c,  which  is  also  the  number  of  the  terms  in  the  quotient  ;• 
and  therefore  the  fluent  is 

xcn—n  unxcn—2n  a2nXrn—3n  q(i — 2 Jr>xi  1 

i ± zz  - a(c-,>n  X 

~ ■ n 


cn—n  cn—2n  cn—3n  n 

hyp.  log.  of  xn  -f-  an.  In  like  manner, 

has  all  the  same 


Ex.  32.  The  fluent  of  — " lx 


xn  _(jn 


terms 


as  the  former,  but  their  signs  all  + or  positive,  and  the  re*  j 


mainder  ~a(c~l^n  X hyp.  log.  of  xn  — an.  Also  in  like  manner, 


n 

Ex.  33. 


The  fluent  of 

an 


has  all  the  very  same  terms, 


but  all  negative,  and  the  remainder  — - a^c~: 0"  X hyp.  log. 

71 

of  aD  — xn. 

j d h 

e b 


Ex.  34.  The  fluent  of ' 


-,cxn 


same  with  the  preceding,  by  substitut.  - for  an,  and  multiply- 


ing the  whole  series  by  the  fraction  -. 


74.  When  the  numerator  is  compound,  as  well  as  the  de- 
nominator, the  expression  may,  in  a similar  manner  by  divi- 
sion, be  reduced  to  like  terms  admitting  of  finite  fluents: 
Thus  for, 

' Ex.  35.  To  find  the  fluent  of  — r-r~  Xxi  =-a 


cEdx2 


c+d*2 


By 


FLUXIONS  AND  FLUENTS. 


345 


and  its 


By  division  this  becomes  — -xi-t-0^-^  X — — — ; 

J d dd  c , , 

fluent  - ^x2+a^r-  X hyp-  Iog-  °f  2 + x2* 

75.  There  are  certain  methods  of  finding  fluents  one  from 
another,  or  of  deducing  the  fluent  of  a proposed  fluxion  from 
another  fluent  previously  known  or  found.  There  are  hardly 
any  general  rules  however  that  will  suit  all  cases  ; but  they 
mostly  consist  in  assuming  some  quantity  y in  the  form  of  a 
rectangle  or  product  of  two  factors,  which  are  such,  that  the 
one  of  them  drawn  into  the  fluxion  of  the  other  may  be  of  the 
form  of  the  proposed  fluxion  ; then  taking  the  fluxion  of  the 
assumed  rectangle,  there  will  thence  be  deduced  a value  of 
the  proposed  fluxion  in  terms  that  will  often  admit  of  finite 
fluents.  The  manner  in  such  cases  will  better  appear  from 
the  following  examples. 

Ex.  1.  To  find  the  fluent  of 


v''(*2+32) 

Here  it  is  obvious  that  if  y be  assumed  = x (x2  + a2), 
then  one  part  of  the  fluxion  of  this  product,  viz.  x X flux, 
of  y/  (x2  -f-  a2),  will  be  of  the  same  form  as  the  fluxion  pro- 
posed. Putting  theref.  the  assumed  rectangle  y=x  \ / (x2+a2) 

nto  fluxions,  it  is  y =x  */  (x2  +a2 ) + — x2-x . But  as  the 

’o riper  part,  viz.  x ^ / (x2+a2),  does  not  agree  with  any  of 
>ur  preceding  forms,  which  have  been  integrated,  multiply  it 
)y  ^ / (x2-|-a2),  and  subscribe  the  same  as  a denominator  to 
he  product,  by  which  that  part  becomes 

rv  ~x  = X f * - ; this  united  with  the  former  part 

/C«2  + a2)  vO^+a2)  r 

aakes  the  whole  y — 


ai- 


luxion  • 


X2j 


v'(x2+a2)  v/(x2-J-a2) 

\y  — X 


v'Cx2-!-^) 

herefore  Xy—Xa2  Xf r 

•S  (x2  ~f" a'2 ) 

yp.  log.  of  x + y/  (x2  -f*  a2),  by  the  12th  form  of  fluents. 


; hence  the  given 

. J and  its  fluent  is 
v(x2  +a2) 

ixy  ( * 2 -f-a2)  _|a2  X 


Ex.  2 In  like  manner  the  fluent  of 


X2j 


will  be 


>und  from  that  of 


✓ (x2  -f-£2) 
by  the  same  12th  form,  and 


v/(x2-a2) 

— \xy/  (a:2 — a2)  + \a2  X hyp.  log.  x -f-  (x2  —a2). 

Ex.  3.  Also  in  a similar  manner,  bv  the  13th  form,  the 
Vol.  II.  45  fluent 


348 


FLUXIONS  AND  FLUENTS. 


fluent  of 


V (“2  -*0 
comes  out  — ±x  (a2 
sine  x. 


— - will  be  found  from  that  of  - 


and 


j(az—x  2): 

■ x-)  + 4a  X cir.  arc  to  radius  a and 


Ex.  4.  In  like  manner,  the  fluent  of 


x4i 


will  be 


found  from  that  of 


X2x 


Here  it  is  manifest  that  y 


v-Crz-f  a2)' 

must  be  assumed  «=  jr3v/(cr24-a2),  in  order  that  one  part  of  \ 
its  fluxion,  viz.  x X flux  of \/(x2 -\-a2)  may  agree  with  the 
proposed  fluxion.  Thus,  by  taking  the  fluxion,  and  re- 


ducing as  before,  the  fluent  of 


x4j 


V^x2  -fa2) 


- will  be  found  = 


[- x 3 y/(x2  -f  a2)  — 3a2  X /- 


3/  (X2  -f  a2  )■ 

Ex.  5.  Thus  also  the  fluent  of — — -isix3>/(a:2 — a2) 

v'fxa  — a2)  4 v v J 

-M*2  X/ 


X2- 


y/{  *2-a2) 


Ex.  6.  And  the/ 


XU 


✓(a2 


, is  — ix3  (a2  — x2)  -f 


-?a3  X/ 


x2  j 


4 J 3/ {a2  — x2  ) 

In  like  manner  the  student  may  find  the  fluents  of 

X’6i  Xs*  „ , xnjr 


r,&C.tO- 


, where  n is  any  eve*  ? 


v'fxs^u2)  3/(x2&a2y  V(x2±a2)’  _ 

number,  each  from  the  fluent  of  that  which  immediately  < 
precedes  it  in  the  series,  by  substituting  for  y as  before,  n 

Thus  the  fluent  of  —rr^,  = — xn_1 3/  (x2  -}-  a2)  — 


(x2  -j-a2 ) n w ' ’ n 

76.  In  like  manner  we  may  proceed  for  the  series  of  simi- 
lar expressions  where  the  index  of  the  power  of  x in  the  nu- 
merator is  some  odd  number. 

Ex.  1.  To  find  the  fluent  of  — Here  assuming 


V(X2  -ha2) 

y — x 2 3/  (x2-f-a2),  and  taking  the  fluxion,  one  part  of  it 
will  be  similar  to  the  fluxion  proposed.  Thus,  y — 2xi 

(x2  + a2)  “I . ; hence  at  once  the  given  fluxion 

Jr3  r — y — 2xxv/(x2-j-a2)  ; theref.  the  required  fluent 


V/(x'2  ■*-u2) 

is  y - f.  2x'x  (x2-fa2)  = x2  3/  (x2-fa2)— § (x2+a3)2. 


by  the  2d  form  of  fluents 


Ex.  2 


FLUXIONS  AND  FLUENTS. 


SV, 


Ex.  2.  In  like  manner  the  fluent  of 


X"i  ; 


v/(*2  -f-ns)’ 


x2  y/  (ar2  — a2)  — f(a?2  — a2)2. 
Ex.  3.  And  the  fluent  of 


f 3 L 


V(a 


rrafe  = -**  ✓(•*-**) 


-i(“s  -*»)«. 

Ex.  4.  To  find  the  flu.  of- 


r,  from  that  of- 


xs: 


v'C^+aa)  ' V(-*2+«2)" 

Here  it  is  manifest  we  must  assume  y = x4  y/  ( x 2 + a2). 
This  in  fluxions  and  reduced  gives  y — 5x5x  ' ‘iu~x3-v 


and  hence 


X$x 


■/(*2+«2) 


— iy 


4 a2 

~r~ 


y/(*2+a  S) 
X3x 

-J  (*2  +32  ) 


v/(x2  -f-a2)5 

; and  the  flu. 


is 


r)=4xV(*3+«aWo3  X/- 


x2., 


^(x2+a2>)  5 v\  /a  •/^(*2+a2) 

the  fluent  of  the  latter  part  being  as  in  ex  1,  above. 

In  like  manner  the  student  may  find  the  fluents  of 

— — — - and  — ,X  -- — . He  may  then  proceed  in  a similar 
a2)  S{a2~  x2)  J 1 

way  for  the  fluents  of  — x‘.~ — , — ~~ x — - &c.  — — — * 

J %/(x2d;a2  )’  S(X2  ±a2),  ^(22*02)’ 

where  n is  any  odd  number,  viz.  always  by  means  of  the 
fluent  of  each  preceding  term  in  the  series 

77.  In  a similar  manner  may  the  process  be  for  the  fluents 
of  the  series  of  fluxions, 

x Xx a?2  x , « 3cn x 

</(«  £*)’  ✓ («  &XY  V(.n&xV  * ' 'vW®)’ 
rising  the  fluent  of  each  preceding  term  in  the  series,  as  a 
part  of  the  next  term,  and  knowing  that  the  fluent  of  the 

first  term 


■ya- 


is  given,  by  the  2d  form  of  fluents,  — 
2 y/  (a  + a;) , of  the  same  sign  as  x. 

Ex.  I.  To  find  the  fluent  of  — — — , having  given  that 

= 2 y/  (x-f-a)  = a suppose.  Here  it  is  evident 


of  x 
v' (x-f-a) 

we  must  assume  y 


$Xx 


x y/  (.x-j-a),  for  then  its  flux,  y ■■ — 

^ 3 /(*+«) 

...  . . ixx  , x‘x  , ax  2Xx  . 

X\/\X  a)  </(x-t-a)  ^(x-f-j)  ✓ (x+a)  ®A  ’ 

hence = tl/  ~ ia‘A  > and  the  required  fluent  is  § y— 

|oa  =| Xy/(x  + a)  — f ay/  (x+a)  = (x  — 2a)  X f y/  (x-f-a). 
In  like  manner  the  student  will  find  the  fluents  of 
xi  , Xx 


and 


y/(x—a)  y (« — x) 


Ex.  2 


348 


FLUXIONS  AND  FLUENTS. 


Ex.  2.  To  find  the  fluent  of- 


haviDg  given  that 


of  — - 
✓(*+«) 


Here  y must  be  assumed  — i - (x-fa)  ; 

X2‘ 


for  then  taking  the  flu.  and  reducing,  there  is  found 


%y—  las;  theref./- 


X^- 


V(x+a) 


= h—  iaB  = f*2  v/  {X  + a) 


•y  (^+a) 

— f ob  — I*2  v/  {x  + a>  - |a  (r  — 2a)  X|v'(r  + a)  = 
(9x2  — 4ax  -{-  8a2)  X t2j  (x  -f-  a)» 

In  the  same  manner  the  student  will  find  the  fluents  of 

X2X 


~;X  x — - and  of  — 7---  --.  And  in  general,  the  fluent  of  — — — 
V(x-r)  v(o-*)  8 v/(*+c) 


2 

Ew+T 


being  given  = c,  he  will  find  the  fluent  of  — — — 

v'tx+a) 

x ./  (x  -f-  a) ac. 

78.  In  a similar  way  we  might  proceed  to  find  the 
fluents  of  other  classes  of  fluxions  by  means . of  other . 
fluents  in  the  table  of  forms  ; as,  for  instance,  such  as 
xx^/(dx—x2),x2x^/(dx — x2),x3Xy/(dx — x2),  k. c.  depend- 
ing on  the  fluent  of. x^/(dx — x2),  the  fluent  of  which,  by 
the  16th  tabular  form,  is  the  circular  semisegment  to  diame- 
ter d and  versed  sine  x , or  the  half  or  trilineal  segment  con- 
tained by  an  arc  with  its  right  sine,  and  versed  sine,  the  diame- 
ter being  d. 

Ex.  1.  Putting  then  said  semiseg.  or  fluent  of  x V(dx-x2) 
= a,  to  find  the  fluent  of  xXx/(dx — x2.  Here  assuming 

.3 

y = (dx  — x2)2,  and  taking  the  fluxions,  they  are  y — 
f (dx  — 2xx)  ^ / (dx  — x2)  ; hence  xx  y/  (dx  — x2)  = \dxy/ 
(dx  — x2)  — iy  = xd\  — %y  ; therefore  the  required  fluent, 

. “ 3 

fxXx/(dx — x2),  is  \dA  — ±y  = \dx  — i(dx  — x2)2=b  suppose. 

Ex.  2.  To  find  the  fluent  of  x2Xy/(dx — x2),  having  that 
of  xXy/  (dx — x2)  given  = b Here  assuming y=x  (dx — x2), 
then  taking  the  fluxions,  and  reducing,  there  results  y = 
(fdx.r—  4x2x)xs  (dx — x2)  ; hence  x2xv/  (dx — x3)  = ^dxx 
■\/(dx — x 2 ) — Xy  =|dB'  — \y  ,the  flu.theref.  of  x2x^/(dx — x2) 


is  fdB — \y  = |dB — |x(dx — x5)-. 


Ex.  3.  In  the  same  manner  the  series  may  be  continued 
to  any  extent  ; so  that  in  general,  the  flu.  of  xn~^x/(dx— x2) 
being  given  = c,  then  the  next,  or  the  flu.  of  x"x^/(dx — xz) 


will  be-"'*"1  — l-dc i-  xn_I  (dx — xz)2. 

n- f-2  - n + 2_ 


79.  To  find  the  fluent  of  such  expressions  as 


a case  not  included  in  the  table  of  forms. 


a/(x2 


-2  ax)> 
Put 


FLUXIONS  AND  FLUENTS. 


349 


Pat  the  proposed  radical  y/  ( x 2 ± 2 ax)  = z,  or  x2  ± 2ax 
— z 2 ; IheD,  completing  the  square,  x2  ±2dx+a2=22-f-  a2, 
and  the  root  isx  ± a = ^(x2  + a2)-  The  fluxion  of  this  is 

* y/  (z*  -fa2)  ’ merel  y/  (X*  ± 2ax)  y/  (z2  + a2)  ’ lne 
fluent  of  which,  by  the  12th  form,  is  the  hyp.  log.  of  z -{-  y/ 
(z2+a2)  = hyp.  log.  of  x di  a + (x2  dt  2 ax),  the  fluent 

required. 


-Ex.  2.  To  find  now  the  fluent  of 


CC x 


\/(x2  -f  2ax) 


, having 


given,  by  the  above  example,  the  fluent  of  — — E ^ = a 
suppose.  Assume  y/  (x2  -f  2ax)  = y ; then  its  fluxion  is 

vT*2  -f  2ax)  ~ y ’ there  ' y/[x*  + 2 ax)  y ~ s/(x?  f 2ax)  ~ y 
• — a a ; the  fluent  of  which  is  y — ax  = y/  (x2  -f-  2ax)— aA, 
the  fluent  sought. 

Ex.  3.  Thus  also,  this  fluent  of 


| the  flu.  of  the  next  in  the  series,  or 


v/(x2+  2 ax) 
X2x 


being  given, 


will  be  found. 


^(x2  + 2 ax) 

by  assuming  Xy/  (x2  -f  2ax)  = y ; and  so  on  for  any  other 

*WJ—  f ' 

- be  given 


of  the  same  form.  As,  if  the  fluent  of  , „ . 

v'(x24*2ax) 

= c ; then,  by  assuming  xn_1  y/  (x2  -f-  2ax)  = y,  the  fluent 

xnx  1 , , „ . „ x 2n  — 1 

=-x”_1v/  (x2  -f  2 ax)  — 


of 


\/  (x2  + 2ax)  n 
Ex.  4.  In  like  manner,  the  fluent  of 


-ac. 


be  ins; 


y/{xs  — 2 ax) 

I • . , Xx 

given,  as  in  the  first  example,  that  of  ^ — — may  be 

■ found  ; and  thus  the  series  ma^ybe  continued  exactly  as  in 
the  3d  ex.  only  taking  - 2 ax  for  -f-  2ax. 


80.  Again,  having  given  the  fluent  of 


v/(2  ax—xc)' 


which 


is  — X circular  arc  to  radius  a and  versed  sine  x,  the  fluent? 
a 


of 


X 2x  „ 

&c.  . 


may  be 


»/(2ax  — x2)’  y/{2ax~.x2)’  ~ ’ y/(2ax — x2)’ 

assigned  by  the  same  method  of  continuation.  Thus, 

Ex.  1.  For  the  fluent  of — - — — , assumev/(2ax — x2) 

C2ax  ■■  cc*1 ) 

= y ; the  required  fluent  will  be  found  = — y/  (2ax— x2)4-a 
or  arc  to  radius  a and  vers.  x. 

Ex.  2.  In  like  manner  the  fluent  of  — - is 

V (2ax_x2) 


350 


FLUXIONS  AND  FLUENTS, 


where  a denotes  the  arc  mentioned  in  the  last  example. 

Ex.  3.  And  in  general  the  fluent  of  r is 

b */(2ax  - xi ) 

°r?  — 1 1 

ac xn~l  *./  (2ax  — x2)i  where  c is  the  fluent  of 

n 7i  v \ / 

Xn — *ar 

——  - — , the  next  preceding  term  in  the  series. 

4/  I a*CLX  X j 

81.  Thus  also,  the  fluent  of  x J (x — a)  being  given,  = 

3 

§(x  — a)2,  by  the  2d  form,  the  fluents  of  x'x  */  (x  — a), 
x2x^/(x  -a),  &c.  . . x*x  ^ / {x  — a),  may  be  found  And  in 
general,  if  the  fluent  of  xn-’x^/(x  — c)  = c be  given  ; then 

3_ 

by  assuming  x"(x  — a)2  — y,  the  fluent  of  xnx  ^ (x—a)  is 

_ , 2 s . 2na 

found  = 2^**(x  -«)-+ST-3  c. 


82.  Also,  given  the  fluent  of  (a;  — a)mx  which  is 


7)1  -j-  1 

( x — a)"1*1  by  the  2d  form,  the  fluents  of  the  series  (x-a)mxx, 
(x — a)mx~x  &c.  . . . (ar — a)mxnx  can  be  found.  And  in  ge- 
neral, the  fluent  of  ( x — a)mxn—lx  being  given  — c : then 
by  assuming  (ar — a)mt]xn  — y,  the  fluent  of  ( x — a)mxnx  is 

found  = gtf-^"n+3, 

m -f-  n -J-  1 

Also,  by  the  same  way  of  continuation,  the  fluents  of 
xnxy/ia^x)  and  of  xnx  (a£,x)m  maybe  found. 


83.  When  the  fluxional  expression  contains  a trinomial 
quantity,  as  (6  + cx  + a:2),  this  may  be  reduced  to  a bi- 
nomial, by  substituting  another  letter  for  the  unknown  one 
x,  connected  with  half  the  coefficient  of  the  middle  term 
with  its  sign.  Thus,  put  | = x-\-\c  : then  z2  = i2-}-ci+vC:!  ; 
theref.  z2  — \t2  = ar2  + car,  and  z 2 +6  — \c2  = x2  + car+i 
the  given  trinomial  which  is  = z2  -f-  a2,  by  putting  a2  = 
b — Ac2. 


Ex.  1.  To  lind  the  fluent  of  — ; — - — . 

Here  z — x -j-  2 ; then  r2  = ,r2  + 4x  + 4,  and  z*  + 1 = 
5 + 4ar  -j-  x2,  also  x = z ; theref.  the  proposed  fluxion 
3* 

reduces  to 1 — ; the  fluent  of  which,  by  the  1 2th  form  in 

v/Cl+z*)  ’ J 

this  vol.  is  3 hyp.  log  of  z -f-  a/(T  4-  z)  = 3 hyp.  log.  x +2 
4-  -v/  (5  + 4x  + a;*). 

Ex.  2. 


MAXIMA  AND  MINIMA. 


351 


Ex.  2.  To  find  the  fluent  of  x (6  + cx  4*  dx2)  =xx/dX, 

Here  assuming  x -f*  ^ = z ; then  x — z,  and  the  proposed 
flux,  reduces  to  z^/dX</(z2  + j—~-)~zx/dXv/(z2+a2'), 

b c • 

putting  a2  for— ; and  the  fluent  will  be  found  by  a sim- 

ilar process  to  that  employed  in  ex.  1 art.  75. 

Ex.  3.  In  like  manner,  for  the  flu.  of  xn~lx  */  (b  4*  4~ 

* c . i . 

dx2n),  assuming  xn  4-^  — z,  nxn~1x  = z,  and  xn~lx  — - z ; 

hence  x2n  4 - 4 rr  — z2,  and  ( dx2n  4-  cxn  + b)  == 

a 4 (12  . 

(*»•  + -d  * + \)  vAZXv/  (*3  4-  -d ~ =)  y/  d 

b c°~ 

X ^ / ( z 2 ± a2),  putting  ± a3  = — — — ; hence  the  given 

fluxion  becomes  — z */  <2  X »/  (z2±a2),  and  its  fluent  as  in 
n 


the  last  example. 

2QTI — \x 

Ex.  4.  Also,  for  the  fluent  of  — — : assume 

6-f-cx-f-dx2 

xn  4-  — = z,  then  the  fluxion  may  be  reduced  to  the  form 
1 2d  J 

— X — , and  the  fluent  found  as  before. 

(In  x2±a  2 

So  far  on  this  subject  may  suffice  on  the  present  occasion. 
But  the  student  who  may  wish  to  see  more  on  this  branch, 
may  profitably  consult  Mr.  Dealtry’s  very  methodical  and 
ingenious  treatise  on  Fluxions,  lately  published,  from  which 
several  of  the  foregoing  cases  and  examples  have  been  taken 
or  imitated. 


>♦< 


OF  MAXIMA  AND  MINIMA  ; OR,  THE  GREATEST 
AND  LEAST  MAGNITUDE  OF  VARIABLE  OR  FLOW- 
ING QUANTITIES. 

84.  Maximum,  denotes  the  greatest  state  or  quantity  attain- 
able in  any  given  case,  or  the  greatest  value  of  a variable 
quantity  : by  which  it  stands  opposed  to  Minimum,  which  is 
the  least  possible  quantity  in  any  case. 

Thus, 


352 


FLUXIONS. 


Thus  the  expression  or  sum  a 2 + bx,  evidently  increases 
as  x,  or  the  term  bx,  increases  : therefore  the  given  expres- 
sion will  be  the  greatest,  or  a maximum,  when  x is  the  greatest, 
or  infinite  : and  tfie  same  expression  will  be  a minimum,  or 
the  least,  when  x is  the  least,  or  nothing.  . 

Again  in  the  algebraic  expression  a2—bx,  where  a and  b 
denote  constant  or  invariable  quantities,  and  x a flowing  or 
variable  one.  Now,  it  is  evident  that  the  value  of  this  re- 
mainder or  difference,  a 2 —bx,  will  increase,  as  the  term  bx 
or  as  x,  decreases  ; therefore  the  former  will  be  the  greatest, 
when  the  latter  is  the  smallest  ; that  is  8?  — bx  is  a maxi- 
mum, when  x is  the  least,  or  nothing  at  all  ; and  the  differ- 
ence is  the  least,  when  x is  the  greatest. 

85.  Some  variable  quantities  increase  continually  ; and  so 
have  no  maximum,  but  what  is  iniinite.  Others  again  de- 
crease continually  ; and  so  have  no  minimum,  but  what  is  of 
no  magnitude,  or  nothing.  But,  on  the  ether  hand,  some  va- 
riable quantities  increase  only  to  a certain  finite  magnitude, 
called  their  Maximum,  or  greatest  state  and  after  that  they 
decrease  again.  While  others  decrease  to  a certain  finite 
magnitude,  called  their  Minimum,  or  least  state,  and  after- 
wards increase  again.  And  lastly,  some  quantities  have  sev- 
eral maxima  nnd  minima. 


Thus,  for  example,  the  ordinate  bc  of  the  parabola,  or 
such-like  curve,  flowing  along  the  axis  ab  from  the  vertex  a, 
continually  increases,  and  has  no  limit  or  maximum.  And  the 
ordinate  gf  of  the  curve  efh,  flowing  from  e towards  h,  con- 
tinually decreases  to  nothing  when  it  arrives  at  the  point  h. 
But  in  the  circle  ilm,  the  ordinate  only  increases  to  a certain 
magnitude,  namely,  the  radius,  when  it  arrives  at  the  middle 
as  at  ke,  which  is  its  maximum  ; and  alter  that  it  decreases 
again  to  nothing,  at  the  point  m.  And  in  the  curve  noq,  the 
ordinate  decreases  only  to  the  position  op,  where  it  is  least, 
or  a minimum  ; and  after  that  it  continually  increases  towards 
But  in  the  curve  rsu  &c.  the  ordinates  have  several  maxi- 
ma, as  st,  wx,  and  several  minima,  as  vu,  vz,  &c. 


51.  Now 


MAXIMA  AND  MINIMA. 


353 


86.  Now,  because  the  fluxion  of  a variable  quantity,  is 
the  rate  of  its  increase  or  decrease  : and  because  the  max- 
imum or  minimum  of  a quantity  neither  increases  nor  de- 
creases, at  those  points  or  states  ; therefore  such  maximum 
or  minimum  has  no  fluxion,  or  the  fluxion  is  then  equal  to  noth- 
ing. From  which  we  have  the  following  rule. 

To  find  the  Maximum  or  Minimum. 

87.  From  the  nature  of  the  question  or  problem,  find  an 
algebraical  expression  for  the  value,  or  general  state  of  the 
quantity  whose  maximum  or  minimum  is  required  ; then 
take  the  fluxion  of  that  expression,  and  put  it  equal  to  no- 
thing ; from  which  equation,  by  dividing  by,  or  leaving  out, 
the  fluxional  letter  and  other  common  quantities,  and  perform- 
ing other  proper  reductions,  as  in  common  algebra,  the  value 
of  the  unknown  quantity  will  be  obtained,  determining  the 
point  of  the  maximum  or  minimum. 

| ' 

So,  if  it  be  required  to  find  the  maximum  state  of  the 
t compound  expression  lQCbr  — 5x3  ± c,  or  the  value  of  x 
when  lOOx  — 5x2±  c is  a maximum.  The  fluxion  of  this 
expression  is  100j?  — lOxj;  — 0 ; which  being  made  = 0, 
and  divided  by  10^,  the  equation  is  10  — x = 0 ; and  hence 
x — 10.  That  is,  the  value  of  a;  is  10,  when  the  expression 
100.r  — 5a: 2 ± c is  the  greatest.  As  is  easily  tried  : for  if  10 
be  substituted  for  x,  in  that  expression,  it  becomes  ± c-|-500  : 
i but  if,  for  x there  be  substituted  any  other  number,  whether 
greater  or  less  than  10,  that  expression  will  always  be  found 
to  be  less  than  ± c -f-  500,  which  is  therefore  its  greatest 
possible  value,  as  its  maximum. 

88  It  is  evident,  that  if  a maximum  or  minimum  be  any 
way  compounded  with,  or  operated  on,-  by  a given  constant 
quantity,  the  result  will  still  be  a maximum  or  minimum. 
That  is,  if  a maximum  or  minimum  be  increased,  or  de- 
creased, or  multiplied,  or  divided,  by  a given  quantity,  or 
any  given  power  or  root  of  it  be  taken  ; the  result  will  still 
be  a maximum  or  minimum.  Thus,  if  x be  a maximum  or 

x 

minimum,  then  also  is  x -j-  a,  or  x — a,  or  or,  or — , or  xa, 

a 

or  ^/x,  still  a maximum  or  minimum.  Also,  the  logarithm 
of  the  same  will  be  a maximum  or  a minimum.  And  there- 
fore, if  any  proposed  maximum  or  minimum  can  be  made 
simpler  by  performing  any  of  these  operations,  it  is 'better  to 
lo  so,  before  the  expression  is  put  into  fluxions, 
i Vol.  II.  46 


89.  When 


354 


FLUXIONS. 


89.  When  the  expression  for  a maximum  or  minimum  con- 
tains several  variable  letters  or  quantities  ; take  the  fluxion 
of  it  as  often  as  there  are  variable  letters  ; supposing  first  one 
of  them  only  to  flow,  and  the  rest  to  be  constant  ; then  an- 
other only  to  flow,  and  the  rest  constant  ; and  so  on  for  all  of 
them  : then  putting  each  of  these  fluxions  = 0,  there  will  be 
as  many  equations  as  unknown  letters,  from  which  these  may 
be  all  determined.  For  the  fluxion  of  the  expression  must  be 
equal  to  nothing  in  each  of  these  cases  ; otherwise  the  ex- 
pression might  become  greater  or  less,  without  altering  the 
values  of  the  other  letters,  which  are  considered  as  constant. 

So,  if  it  be  required  to  find  the  values  of  x and  y when 
lx2  — xy  + 2y  is  a minimum.  Then  we  have, 

First,  - 8xx  — xy  — 0,  and  8x—  y = 0,  or  y = 8#. 

Secondly,  2y  — xy  = 0,  and  2 — x = 0,  or  x — 2. 

And  hence  y or  8x  — 16. 

90.  To  find  mh  ether  a proposed  quantity  admits  of  a Maxi- 
. mum  or  a Minimum. 

Every  algebraic  expression  does  not  admit  of  a maximum 
or  minimum,  properly  so  called  ; for  it  may  either  increase 
continually  to  infinity,  or  decrease  continually  to  nothing  ; 
and  in  both  these  cases  there  is  neither  a proper  maximum 
nor  minimum  ; for  the  true  maximum  is  that  finite  value  to 
which  an  expression  increases,  and  after  which  it  decreases 
again  : and  the  minimum  is  that  finite  value  to  which  the  ex- 
pression decreases,  and  alter  that  it  increases  again.  There- 
fore, W'hen  the  expression  admits  of  a maximum,  its  fluxion  is 
positiie  before  the  point,  and  negative  after  it ; but  when  it  ad- 
mits of  a minimum,  its  fluxion  is  negative  before,  and  positive 
after  it.  Hence  then,  taking  the  fluxion  of  the  expression  a 
little  before  the  fluxion  is  equal  to  nothing,  and  again  a little 
after  the  same  ; if  the  former  fluxion  be  positive,  and  the  lat- 
ter negative,  the  middle  state  is  a maximum,  but  if  the  former 
fluxion  be  negative,  and  the  latter  positive,  the  middle  state  is 
minimum. 

So  if  we  would  find  the  quantity  ax — x 2 a maximum  or 
minimum  ; make  its  fluxion  equal  to  nothing,  that  is, 
ax-  2xx  = 0,  or  (a  — 2x)x  = 0 ; dividing  by  x,  gives 
a — 2x  = 0,  or  x = £a  at  that  state.  Now,  if  in  the-fluxion 
(o — 2x)  x,  the  value  of  x be  taken  rather  less  than  its  true 
value,  iff,  that  fluxion  will  evidently  be  positive  ; but  if  x be 
taken  somewhat  greater  tbau  the  value  of  a — 2x,  anfl 
consequently  of  the  fluxion,  is  as  evidently  negative.  There- 
fore, the  fluxion  of  ax  — x 2 being  positive  before,  and  ne- 
gative 


MAXIMA  AND  MINIMA. 


355 


gative  after  the  state  when  its  fluxion  is  = 0,  it  follows  that 
at  this  state  the  expression  is  not  a minimum  but  a maximum. 

Again,  taking  the  expression  x3 — ax2,  its  fluxion  3ir2  x — 
2axx—{3x — 2a)xx— 0 ; this  divided  bv  xx  gives  3x — la— 0, 
and  x — | a its  true  value  when  the  fluxion  of  x3  — ax2  is 
equal  to  nothing.  But  now  to  know  whether  the  given  ex- 
pression be  a maximum  or  a minimum  at  that  time,  take  x a 
little  less  than  in  the  value  of  the  fluxion  (3a:  — 2a)  x'x, 
and  this  will  evidently  be  negative  ; and  again,  taking  x a 
little  more  than  f a,  the  value  of  3x  — 2a  or  of  the  fluxion, 
is  as  evidently  positive  Therefore  the  fluxion  of  r3  — ax2 
being  negative  before  that  fluxion  is  = 0,  and  positive  after 
it,  it  follows  that  in  this  state  the  quantity  a:3  — ax2  admits 
of  a minimum,  but  not  of  a maximum. 

91.  SOME  EXAMPLES  FOR  PRACTICE. 

Exam  1 To  divide  a line,  or  any  other  given  quantity  a, 
into  two  parts,  so  that  their  rectangle  or  product  may  be  the 
greatest  possible. 

Exam.  2.  To  ‘divide  the  given  quantity  a into  two  parts 
such  that  the  product  of  the  in  power  of  one,  by  the  n 
power  of  the  other,  may  be  a maximum. 

Exam  3 To  divide  the  given  quantity  a into  three  parts 
such  that  the  continual  product  of  them  all  may  he  a 
maximum. 

Exam.  4.  To  divide  the  given  quantity  a into  three  parts 
| such,  that  the  coutinual  product  of  the  1st,  the  square  of  the 
2d,  and  the  cube  of  the  3d,  ma}'  be  a maximum. 

Exam  5.  To  determine  a fraction  such  that  the  difl’er- 
! ence  between  its  m power  and  n power  shall  be  the  greatest 
1 possible- 

Exam  6.  To  divide  the  number  CO  into  two  such  parts 
I x and  y,  that  2x2  -f - xy  3 y2  may  be  a minimum. 

Exam.  7.  To  find  the  greatest  rectangle  that  can  be  in- 

1>  scribed  in  a given  right-angled  triangle. 

Exam.  8.  To  find  the  greatest  rectangle  that  can  be  inscri- 
bed in  the  quadrant  of  a given  circle. 

I Exam.  9.  To  find  the  least  right-angled  triangle  that  can 
circumscribe  the  quadrant  of  a given  circle. 

Exam.  10.  To  find  the  greatest  rectangle  inscribed  in,  and 
\ the  least  isosceles  triangle  circumscribed  about,  a given  semi- 
ellipse. 


Exam.  1 1 . 


356 


TANGENTS. 


Exam.  11.  To  determine  the  same  for  a given  parabola 

Exam.  12.  To  determine  the  same  for  a given  hyperbola. 

Exam.  13.  To  inscribe  the  greatest  cylinder  in  a given 
cone  ; or  to  cut  the  greatest  cylinder  out  of  a given  cone. 

Exam.  14.  To  determine  the  dimensions  of  a rectangular 
cistern,  capable  of  containing  a given  quantity  a of  water,  so 
as  to  be  lined  with  lead  at  the  least  possible  expense. 

Exam.  15.  Required  the  dimensions  of  a cylindrical  tan- 
kard, to  hold  one  quart  of  ale  measure,  that  can  be  made  of 
the  least  possible  quantity  of  silver,  of  a given  thickness. 

Exam.  16.  To  cut  the  greatest  parabola  from  a given 
cone. 

Exam.  17.  To  cut  the  greatest  ellipse  from  a given  cone. 

Exam.  18.  To  find  the  value  of  x when  xx  is  a minimum 


THE  METHOD  OF  TANGENTS  ; OR,  TO  DRAW 
TANGENTS  TO  CURVES. 

92.  The  Method  of  Tangents,  is  a method  of  determining 
the  quantity  of  the  tangent  and  subtangent  of  any  algebraic 
curve  : the  equation  of  the  curve  being  given.  Or,  vice  versa, 
the  nature  of  the  curve,  from  the  tangent  given. 


If  ae  be  any  curve,  and  e be  any 
point  in  it,  to  which  it  is  required 
to  draw  a tangent  te.  Draw  the 
ordinate  ed  : then  if  we  can  deter- 
mine the  subtangent  td,  limited  be- 
tween the  ordinate  and  tangent,  in 
the  axis  produced,  by  joining  the 
poiuts,  t,  e,  the  line  te  will  be  the 
tangent  sought. 


93.  Let  dae  be  another  ordinate,  indefinitely  near  to  de 
meetiug  the  curve,  or  tangent  produced  in  e ; and  let  F.e  be 
parallel  to  the  axis  ad.  Then  is  the  elementary  triangle 
e ea  similar  to  the  triangle  tde  ; and 


therefore 


TANGENTS. 


357 


therefore  - ea  : oe  : : ed  : dt. 

But  - - ea  : as  : : flux,  ed  : flux.  ad. 

Therefore  - flux,  ed  : flux,  ad  : : de  : dt. 

That  is  - - y : a:  ::  y — dt. 

y 

which  is  therefore  the  general  value  of  the  subtangent  sought  ; 
where  x is  the  absciss  ad,  and  y the  ordinate  de. 

Hence  we  have  this  general  rule. 


GENERAL  RULE. 


94.  By  means  of  the  given  equation  of  the  curve,  when 
put  into  fluxions,  find  the  value  of  either  x or  y or  of  ; 


which  value  substitute  for  it  in  the  expression  dt 

y 

I and,  when  reduced  to  its  simplest  terms,  it  will  be  the  value 
| of  the  subtangent  sought. 


EXAMPLES. 

Exam.  1.  Let  the  proposed  curve  be  that  which  is  defined, 
or  expressed  by  the  equation  ax2~r  xy2 — y 3 =0. 

Here  the  fluxion  of  the  equation  of  the  curve  is 
2axx-\-y2 x-^r^xyy — 3y2y  _Q  ; then,  by  transposition, 

~&x x-\-y~ x=3y2 y — 2a: yy  ; and  hence,  by  division, 

* 3y2  — 2xy  . ifx  3y3 — 7xy2 

r~=— ; consequently  ^ = ■■  ■ , — — . 

y 2ax-{-y2  1 J y 2 ax-\-y2 

which  is  the  value  of  the  subtangent  td  sought. 

Exam.  2.  To  draw  a tangent  to  a circle  ; the  equation  of 
which  is  ax  — x2  — y2  ; where  x is  the  absciss,  y the  ordi- 
nate, and  a the  diameter. 

Exam.  3.  To  draw  a tangent  to  a parabola  ; its  equation 
being  ax  = y2  • where  a denotes  the  parameter  of  the  axis. 

Exam.  4.  To  draw  a tangent  to  an  ellipse  ; its  equation 
being  c2  (ax  — x2}  = a2 y2  ; where  a and  c are  the  two  axes. 

Exam.  5.  To  draw  a tangent  to  an  hyperbola;  its  equa- 
tion being  c 2 (ax  + x2)  —a2y2  • where  a and  c are  the  two 
axes. 

Exam.  6.  To  draw  a tangent  to  the  hyperbola  referred  to 
the  asymptote  as  an  axis  ; its  equation  being  xy  =a2  ; where 
a*  denotes  the  rectangle  of  the  absciss  and  ordinate  answer 
?ng  to  the  vertex  of  the  curve. 

OF 


358 


RECTIFICATIONS. 


OF  RECTIFICATIONS  ; OR,  TO  FIND  THE 
LENGTHS  OF  CURVE  LINES. 


95.  Rectification,  is  the  finding  the  length  of  a curve 
line,  or  finding  a right  line  equal  to  a proposed  curve. 

By  art  10  it  appears,  that  the 
elementary  triangle  e ae,  formed  hy 
the  increments  of  the  absciss,  ordinate, 
and  curve,  is  a right-angied  triangle,  of 
which  the  increment  of  the  curve  is 
the  hypothenuse  : and  therefore  the 
square  of  the  latter  is  equal  to  the  sum 
of  the  squares  of  the  two  former  ; that  is,  e e2  = e<i2  -f-  ae2. 
Or,  substituting,  for  the  increments,  their  proportional 
fluxions,  it  is  zz  — xx  + yy  , or  'z  — x2  +y  2 ; where  2 de- 
notes any  curve  line  ae,  x its  absciss  ad,  and  y its  ordinate  de. 
Hence  this  rule. 

RULE. 


96.  From  the  given  equation  of  the  curve  put  into 
fluxions,  find  the  value  of  x2  ar  y2,  which  value  substitute 
instead  of  it  in  the  equation. z — ^/  x2  2 ; then  the  fluents, 
being  taken,  will  give  the  value  of  z,  or  the  length  of  the 
curve,  in  terms  of  the  absciss  or  ordinate. 

EXAMPLES. 


Exam.  I.  To  find  the  length  of  the  arc  of  a circle,  in  terms 
of  the  sine,  versed  sine,  tangent,  and  secant. 

The  equation  of  the  circle  may  be  expressed  in  terms  of 
the  radius,  and  either  the  sine,  or  the  versed  sine,  or  tangent, 
or  secant,  & c.  of  an  arc.  Let  therefore  the  radius  of  the 
circle  be  ca  or  ce  — r,  the  versed  sine  ad  (of  the  arc  ae)  = x 
the  right  sine  de  = y,  the  tangent  te  = t,  and  the  secant 
ct  = s,  then,  by  the  nature  of  the  circle,  there  arise  these 
equations,  viz. 


y2  — 2 rx  ~ x- 


r2  i2  s~  — r2 
r-  ^ : 


Then,  by  means  of  the  fluxions  of  these  equations,  with 
the  general  fluxional  equation  z 3 = x2  + y2 , are  obtained  the 
following  fluxional  forms,  for  the  fluxion  of  the  curve  ; the 
fluent  of  any  cne  of  which  will  be  the  curve  itself  ; viz. 


ry 


v'2  rx—  xx  aJ  r2  — y- 


r2  +12 


a/s2  - r2 


Hence 


FLUXIONS. 


.369 


Hence  the  value  of  the  curve,  from  the  duent  of  each  of 
these,  gives  the  four  following  forms,  in  series,  viz.  putting 
d = 2r  the  diameter,  the  curve  is 


0x2 


z = (\  -f-  — + 

" ' 2.3d  2 AM 

y2  , 3.v4 


+ 


3.5x: 


■ &C.)  dr. 


2.4.6.7ds 

= (1+2Jfc+ §£74 + + &c-)  y> 

t2  t\  Ip  Jf8 

= (1  - ~+  1 — -j-  — &c.)  t, 

' • 3r2  ir4  7>6  9r8-  ' 

_ ,s  — r , ss  — 1-3  3(iS_rS) 

= t—  + nar  + r- 


2.4.5.S5 


Now,  it  is  evident  that  the  simplest  of  these  series,  is  the 
third  in  order,  or  that  which  is  expressed  in  terms  of  the 
tangent.  That  form  will  therefore  be  the  fittest  to  cal- 
culate an  example  by  in  numbers.  And  for  this  purpose  it 
will  be  convenient  to  assume  some  arc  whose  tangent,  or  at 
least  the  square  of  it,  is  known  to  be  some  small  simple 
number.  Now,  the  arc  of  45  degrees,  it  is  known,  has  its 
tangent  equal  to  the  radius  ; and  therefore,  taking  the  radius 
r—  1,  and  consequently  the  tangent  of  45°,  or  t,  = I also, 
in  this  case  the  arc  of  45°  to  the  radius  1,  or  the  arc  of  the 
quadrant  to  the  diameter  1,  will  be  equal  to  the  infinite  series 

1 — i + i — 4 + I - &c- 


But  as  this  series  converges  very  slowly,  it  will  be  proper 
to  take  some  smaller  arc,  that  the  scries  may  converge 
faster  ; such  as  the  arc,  of  30  degrees,  the  tangent  of  which 
is  = ^/i,  or  its  square  t2  — J-  : which  being  substituted  in 
the  series,  the  length  of  the  arc  of  30°  comes  out 

(1 -- -j — i-  - — L— - — &c.)  a/i.  Hence,  to  com- 

v 3 3 5.32  7.33  1 9.5 4 1 v 3 

pute  these  terms  in  decimal  numbers,  after  the  first,  the  suc- 
ceeding  terms  will  be  found  by  dividing,  always  by  3,  and 
these  quotients  again  by  the  absolute  number  3,  5,  7,  9,  &c.  ; 
and  lastly,  adding  every  other  term  together,  into  two  sums, 
the  one  the  sum  of  the  positive  terms,  and  the  other  the  sum 

I of  the  negative  ones  ; then  lastly,  the  one  sum  taken  from 
the  other  leaves  the  length  of  the  arc  of  30  degrees  ; cvhich 
being  the  12th  part  of  the  whole  circumference  when  the 
radius  is  1,  or  the  Gth  part  when  the  diameter  is  1,  conse- 
quently 6 times  that  arc  will  be  the  length  of  the  whole  cir- 
cumference to  the  diameter  1.  Therefore  multiplying  the 
first  term  by  G,  the  product  is  12  = 3-4641016  ; and 
hence  the  operation  will  be  conveniently  made  as  follows  : 


4*  Terms 


360 


QUADRATURES. 


+Terms. 

— Terms. 

1 ) 

3-4641016 

( 

3 4641016 

3) 

1-1547005 

( 

0-3849002 

5 ) 

3849002 

( 

769800 

7) 

1283001 

( 

183286 

9) 

427667 

( 

47519 

11  ) 

142556 

( 

12960 

13) 

47519 

( 

3655 

15) 

15840 

( 

1056 

17) 

5280 

311 

19  ) 

1760 

r 

93 

21  ) 

587 

( 

28 

23  ) 

196 

( 

8 

25  ) 

65 

( 

3 

27  ) 

22 

( 

1 

+3-5462332  —0-4046406 

—0-4046406 


So  that  at  last  3-1415926  is  the  whole  circumfer- 
ence  to  the  diameter  1. 

Exam  2.  To  find  the  length  of  a parabola. 

Exam  3.  To  find  the  length  of  the  semicubical  parabola, 
whose  equation  is  ax2  = y3. 

Exam  4.  To  find  the  length  of  an  elliptical  curve. 

Exam  5.  To  find  the  length  of  an  hyperbolic  curve. 


OF  QUADRATURES  ; OR,  FINDING  THE  AREAS 
OF  CURVES. 

S7.  The  Quadrature  of  Curves,  is  the  measuring  their 
areas,  or  finding  a square,  or  other  right-lined  space,  equal 
to  a nroposed  curvilineal  one. 

By  art.  9 it  appears,  that  any  flowing 
quantity  being'  drawn  into  the  fluxion  of 
the  line  along  which  it  flows,  or  in  the 
direction  of  its  motion,  there  is  produced 
the  fluxion  of  the  quantity  generated  by 
the  flowing.  That  is,  d d X de  or  yx  is 
the  fluxion  of  the  area  ade.  Hence  this 
rule. 


RULE 


SURFACES. 


361 


RULE. 


98.  From  the  given  equation  of  the  curve,  find  the  value 
either  of  x or  of  y ; which  value  substitute  instead  of  it  in 
the  expression  yx  ; then  the  fluent  of  that  expression,  being 
taken,  will  be  the  area  of  the  curve  sought. 


EXAMPLES. 


Exam.  1.  To  find  the  area  of  the  common  parabola. 

The  equation  of  the  parabola  being  ax  = y2  ; where  a is 
the  parameter,  x the  absciss  ad,  or  part  of  the  axis,  and  y 
the  ordinate  de. 

From  the  equation  of  the  curve  is  found  y = ax.  This 
substituted  in  the  general  fluxion  of  the  area  yx  gives  x ax 
\ X 

or  a2x2x  the  fluxion  of  the  parabolic  area  ; and  the  fluent 

- . i 3. 

of  this,  or  fa2x2  = f x y/  ax  = frn/,  is  the  area  of  the  para- 
bola  ade,  and  which  is  therefore  equal  to  § of  its  circum- 
scribing rectangle- 

Exam.  2.  To  square  the  circle,  or  find  its  area. 

The  equation  of  the  circle  being  y2  = ax  — x2,  or  y — 
y/ax—x2,  where  a is  the  diameter;  by  substitution,  the 
general  fluxion  of  the  area  yx,  becomes  x ax—x2,  for  the 
fluxion  of  the  circular  area.  But  as  the  fluent  of  this  cannot 
be  found  in  finite  terms,  the  quantity  ax— a2  is  thrown 
into  a series,  by  extracting  the  root,  and  then  the  fluxion  of 
the  area  becomes 

X 0 -s -sis s5~&c-)  i 

ind  then  the  fluent  of  every  term  being  taken,  it  gives 
_ , 2 1.x  x2  1.3x3  1-3.5.X4 


x y/  ax  X (- 


-&c.)  ; 


3 5a  4.7a2  4.6.9  a3  4.6.8.11a4 

or  the  general  expression  of  the  semisegment  ade. 

And  when  the  point  d arrives  at  the  extremity  of  the  dia- 
meter, then  the  space  becomes  a semicircle,  and  x — a ; and 
hen  the  series  above  becomes  barely 

„,2  1 1 L3  1.3.5.  . 

a2  ( &c. ) 

' 3 5 4 7 4.6.9  4.6.8.11  ' 

or  the  area  of  the  semicircle  whose  diameter  is  a. 
j For,.  If.  • 47 


Exam,  3, 


362 


FLUXIONS'. 


Exam.  3.  To  find  the  area  of  any  parabola,  whose  equation 
is  amzn—ym*n. 

Exam.  4.  To  find  the  area  of  an  ellipse. 

Exam.  5.  To  find  the  area  of  an  hyperbola. 

Exam.  6.  To  find  the  area  between  the  curve  and  asymp- 
tote of  afi  hyperbola. 

Exam.  7 To  find  the  like  area  in  any  other  hyperbola 
whose  general  equation  is  xmyn=amfn. 


TO  FIND  THE  SURFACES  OF  SOLIDS. 

99.  In  the  solid  formed  by  the  rota- 
tion of  any  curve  about  its  axis,  the 
surface  may  be  considered  as  generated 
by  the  ciricumference  of  an  expanding 
circle,  moving  perpendicularly  along 
the  axis,  but  the  expanding  circum- 
ference moving  along  the  arc  or  curve 
ef  the  solid  Therefore,  as  the  fluxion 
of  any  generated  quantity  is  produced  by  drawing  the  ge- 
nerating quantity  into  the  fluxion  of  the  line  or  direction  in 
which  it  moves,  the  fluxion  of  the  surface  will  be  found  by 
drawing  the  circumference  of  the  generating  circle  into  the 
fluxion  of  the  curve,  That  is,  the  fluxion  of  the  surface, 
bae,  is  equal  to  ae  drawn  iuto  the  circumference  ecef,  1 
whose  radius  is  the  ordinate  de. 

100.  But,  if  c be  = 3-1416,  the  circumference  of  a circle 
whose  diameter  is  1,  x = ad  the  absciss,  y = de  the  ordi- 
nate, and  2 = ae  the  curve  ; then  2y  = the  diameter  be, 
apd  2 cy  = the  circumference  bcef  ; also,  ae  — z — 

x + y~  : therefore  Icy-  or  2 cy  x2~h  y2  is  the  fluxion  of 
the  surface.  And  consequently  if,  from  the  given  equation 
of  the  curve,  the  value  of  x or  y be  found,  and  substituted 
in  this  expression  2 cy  x / x2  + y 2 , the  fluent  of  the  expression 
being  then  taken,  will  be  the  surface  of  the  solid  required. 

EXAMPLES. 


Exam.  1.  To  find  the  surface  of  a sphere,  or  of  any  seg- 
ment. 

In 


CUBATURES. 


363 


in  this  case,  ae  is  a circular  arc,  whose  equation  is 


y2  =zax  — x2 , or  y = ax  — x2 . 
The  fluxion  of  this  gives  y = 


c— 2ar  . 
-x 


hence  y2  = 


^■2  ; consequently 


2v/ax-M  ”1/ 

4gJ  -f-  4x2  o2  — 4y2 

4y2  “C  4y2 

■ n i ■„  a2  * 2 j - “ ~ 1 ax 

X-  + y-  ~ -7-^-,  and  Z — -J  X-  + y-  — — ■ 

7 4 y2  v 3 zy 

This,  value  of  z.  the  fluxion  of  a circular  arc,  maybe  found 
more  easily  thus  : In  the  fig.  to  art  95,  the  two  triangles 
edc,  Effe  are  equiangular,  being  each  of  them  equiangular  to 
the  triangle  etc  : conseq  ed  : ec  : : sa  : Ee,  that  is,  - 
ax 


y : \a  : : x : z — — , the  same  as  before. 

2 y 

The  value  of  z being  found,  by  substitution  is  obtained 
2 cy'z  = ac'x  for  the  fluxion  of  the  spherical  surface,  generated 
by  the  circular  arc  in  revolving  about  the  diameter  ad.  And 
the  fluent  of  this  gives  acx  for  the  said  surface  of  the  spheri- 
cal segment  bae. 

But  ac  is  equal  to  the  whole  circumference  of  the  gene- 
rating circle  ; and  therefore  it  follows,  that  the  surface  of 
any  spherical  segment,  is  equal  to  the  same  circumference  of 
the  generating  circle,  drawn  into  x or  ad,  the  height  of  the 


segment. 

Also  when  x or  ad  becomes  equal  to  the  whole  diameter  a, 
i the  expression  acx  becomes  aca  or  ca~ , or  4 times  the  area  of 
the  generating  circle,  for  the  surface  of  the  whole  sphere. 

And  these  agree  with  the  rules  before  found  in  Mensuration 
of  Solids. 


Exam.  2.  To  find  the  surface  of  a spheroid. 
Exam.  3.  To  find  the  surface  of  a paraboloid. 

! Exam.  4.  To  find  the  surface  of  an  hyperboloid 


J 

TO  FIND  THE  CONTENTS  OF  SOLIDS. 

101.  Any  solid  which  is  formed  by  the  revolution  of  a 
curve  about  its  axis  (see  last  fig),  may  also  be  conceived  to 
be  generated  by  the_  motion  of  the  plane  of  an  expanding 
•circle,  moving  perpendicularly  along  the  axis.  And  there” 

fore 


364 


LOGARITHMS 


fore  the  area  of  that  circle  being  drawn  into  the  fluxion  oi 
the  'axis,  will  produce  the  fluxion  of  the  solid.  That  is, 
ad  X area-  of  the  circle  bcf,  whose  radius  is  de,  or  diame- 
ter be,  is  the  fluxion  of  the  solid,  by  art.  9. 

102.  Hence,  if  ad  = x,  de  = y,  c = 3T416  ; because  cy 2 
is  equal  to  the  area  of  the  circle  bcf  : therefore  cy2  i is  the 
fluxion  of  the  solid.  Consequently  if,  from  the  given  equa- 
tion of  the  curve,  the  value  of  either  y-  or  x be  found,  and 
that  value  substituted  for  it  in  the  expression  cy2x,  the  fluent 
of  the  resulting  quantity,  being  taken,  will  be  the  solidity  of 
the  figure  proposed. 

EXAMPLES. 

Exam.  1.  To  find  the  solidity  of  a sphere,  or  any  segment,  . 

The  equation  to  the  generating  circle  being  y2  = ax  — x2 , 
where  a denotes  the  diameter,  by  substitution,  the  general  j 
fluxion  of  the  solid  cy2x,  becomes  caxx—cx2x , the  fluent  of 
which  gives  ±cax2 — %cx3,  or  jfx2  (3a — 2x),  for  the  solid  con- 
tent of  the  spherical  segment  bae,  whose  height  ad  is  x. 

When  the  segment  becomes  equal  to  the  whole  sphere,  ! 
theD  x = a,  and  the  above  expression  for  the  solidity,  be- 
comes ica3  for  the  solid  content  of  the  whole  sphere. 

And  these  deductions  agree  with  the  rules  before  given  and 
demonstrated  in  the  Mensuration  of  Solids. 

Exam  2.  To  find  the  solidity  of  a spheroid. 

Exam,  3.  To  find  the  solidity  of  a paraboloid. 

Exam.  4.  To  find  the  solidity  of  an  hyperboloid. 


TO  FIND  LOGARITHMS. 

108.  It  has  been  proved,  art  23,  that  the  fluxion  of  the 
hyperbolic  logarithm  of  a quantity,  is  equal  to  the 'fluxion  of 
the  quantity  divided  by  the  same  quantity.  Therefore,  when 
anv  quantity  is  proposed,  to  fiud  its  logarithm  ; take  the 
fluxion  of  that  quantity,  and  divide  it  by  the  same  quantity; 
then  take  the  fluent  of  the  quotient,  either  in  a series  or  other- 
wise, and  it  will  be  the  logarithm  sought ; when  corrected  as 
usual,  if  need  be  ; that  is,  the  hyperoolic  logarithm 

104.  But,  for  any  other  logarithm,  multiply  the  hyperbolic 
logarithm,  above  found,  by  the  modulus  of  the  system,  for 
the  logarithm  sought. 


Note. 


FLUXIONS. 


365 


Note.  The  modulus  of  the  hyperbolic  logarithms,  is  1 ; 
and  the  modulus  of  the  common  logarithms,  is  ’43429448190 
&c.  ; and,  in  general,  the  modulus  of  any  system,  is  equal  to 
the  logarithm  of  10  in  that  system  divided  by  the  number 
2-3025850929940  &c.  which  is  the  hyp.  log.  of  10.  Also, 
the  hyp.  log.  of  any  number,  is  in  proportion  to  the  com.  log. 
of  the  same  number,  as  unity  or  1 is  to  ’43429  &c.  or  as  the 
number  2 302585  &c.  is  to  1 ; and  therefore,  if  the  common 
log.  of  any  number  be  multiplied  by  2 302585  &c.  it  will  give 
the  hyp.  log.  of  the  same  number  ; or  if  the  hyp,  log.  be  di- 
vided by  2-302585  &c.  or  multiplied  by  ’43429  &c.  it  will  give 
the  common  logarithm. 


a-f  x 


Exam.  1.  To  find  the  log.  of 

Denoting  any  proposed  number  z , whose-  logarithm  is  re- 
quired to  be  found,  by  the  compound  expression 


-r,  the  fluxion  of  the  number  z,  is  and  the  fluxion 


Xx.  . *X2  X 

a2  a3 


of  the  log.— 

Z II— [- «*/  “ - W - I*  - 

Then  the  fluent  of  these  terms  give  the  logarithm  of 


X 3 Z. 

JL  + &c. 

a* 


or  logarithm  of- 


,a 


x2  a»3 

2^2  +3^' 


x 4 


4a4 


&c. 


n,  ...  i-  • i a—x  x x 2 x3  xi  „ 

Writing  —x  for  x,  gives  log. = — — — — _ — &c. 


a 2a2  3a s 
2x  , 2x3  , 2xs 


i a-4-x  zx  . zjup  „ 

log.  — ■ — = — -f— {- &C. 

a — x a oa3  5as 


Div.  these  numb,  and 
subtr.  their  logs,  gives 

Also,  because  — - = 1 a-=L-,or log.  - 

a±.x  a ® a_ 

p . p a x x2  x3  xi 

therefore  log. of  — ,s-_+— +_+_  &c. 

and  the  log.  of  — is  + - + " «tc. 

a— x a 2a2  3as  4^4 

-*2  1 *4  1 xb  10 

■ TrrT.i h &C. 

a - 2a 4 3a  6 


rv  i 

= 0 — log.— ; 


the  prod,  gives  log. —— — - 


Now,  for  an  example  in  numbers,  suppose  it  were  required 
to  compute  the  common  logarithm  of  the  number  2.  This 
will  be  best  done  by  the  series, 


log.  of 


2 &c. 

n S 'a 5 7/»7 


X* 


a—x 


3as  7« 7 


Making 


I 


INFLECTIONS. 


366 


Making  — 2,  gives  a — 3x  ; conseq.  — = a,  and  ~ 

— i,  which  is  the  constant  factor  for  every  succeeding  term  ; 
also,  2m  — 2 X -43429448190  = -868588964  ; therefore  the 
calculation  will  be  conveniently  made,  by  first  dividing  this 
number  by  3 then  the  quotients  successively  by  9,  and  lastly 
these  quotients  in  order  by  the  respective  numbers  1,  3, 
5,  7,  9,  &c.  and  after  that,  adding  all  the  terms  together,  as 
follows  : 


3 ) 
9 ) 

•868588964 

•289529654 

1 ) 

•289529654 

( 

■289529654 

9 ) 

32169962 

3 ) 

32169962 

( 

10723321 

9 ) 

3574440 

5 ) 

3574440 

( 

f 

714888 

9 ) 

397160 

7 ) 

397160 

56737 

9 ) 

44129 

9 ) 

44129 

( 

4903 

9 ) 

4903 

n ) 

4903 

( 

446 

9 ) 

545 

13  ) 

545 

( 

42 

9 ) 

61 

15  ) 

61 

( 

4 

Sum  of  the  terms  gives  log.  2 = -301029995 


Exam.  2.  To  find  the  log.  of 
Exam.  3.  To  find  the  log.  of  a — x. 
Exam.  4.  To  find  the  log.  of  3. 
Exam.  5.  To  find  the  log.  of  5. 
Exam.  6.  To  find  the  log  of  11. 


TO  FIND  THE  POINTS  OF  INFLEXION,  OR  OF  CON- 
TRARY FLEXURE  IN  CURVES. 


105.  The  Point  of 
Inflexion  in  a curve,  is 
that  point  of  it  which 
separates  the  concaye 
from  the  convex  part, 
lying  between  the  two: 
or  where  the  curve 
changes  from  concave  to  convex,  or  from  convex  to  concave, 
on  the  same  side  of  the  curve.  Such  as  the  point  e in  the 
annexed  figures,  where  the  former  of  the  two  is  concave 

towards 


FLUXIONS, 


367 


towards  the  axis  ad,  from  a to  e,  but  convex  from  e to  f ; and 
on  the  contrary,  the  latter  figure  is  convex  from  a to  e,  and 
and  concave  from  e to  f. 


106.  From  the  nature  of  curvature,  as  has  been  remarked 
before  at  art.  28,  it  is  evident,  that  when  a curve  is  concave 
towards  an  axis,  then  the  fluxion  of  the  ordinate  decreases, 
or  is  in  a decreasing  ratio,  with  regard  to  fluxion  of  the  ab- 
sciss ; but,  on  the  contrary,  that  it  increases,  or  is  in  an  in- 
creasing ratio  to  the  fluxion  of  the  absciss,  when  the  curve  is 
convex  towards  the  axis  ; and  consequently  those  two  fluxions 
are  in  a constant  ratio  at  the  point  of  inflexion,  where  the 
curve  is  neither  convex  nor  concave  ; that  is,  x is  to  y in  a 

constant  ratio,  or  v-  or  - is  a constant  quantity.  But  constant 
* y . . . 

quantities  have  no  fluxion,  or  their  fluxion  is  equal  to  nothing  ; 

so  that  in  this  case,  the  fluxion  of  v—  or  of  ^is  equal  to  nothing. 

x V 

And  hence  we  have  this  general  rule  : 

107.  Put  the  given  equation  of  the  curve  into  fluxions  ; 

from  which  find  either  or  -r.  Then  take  the  fluxion  of  this 
• ■ * y 

ratio,  or  fraction,  and  put  it  equal  to  0 or  nothing  ; and  from 


this  last  equation  find  also  the  value  of  the  same  .-  or  ~ 

y X- 

Then  put  this  latter  value  equal  to  the  former,  which  will 
form  an  equation  ; from  which,  and  the  first  given  equation 
of  the  curve,  x and  y will  be  determined,  being  the  absciss 
and  ordinate  answering  to  the  point  of  inflexion  in  the  curve 
as  required. 


EXAMPLES 


Exam.  1.  To  find  the  point  of  inflexion  in  the  curve 
whose  equation  is  ax 2 = a2y  -j-  x2y. 

This  equation  in  fluxions  is  2 axx  — a2  y -f-  2 xyx  + *2  y. 

which  gives'1  = ^ Then  the  fluxion  of  this  quantiti 

V 2 ax—'2xy  y •> 

made  = 0,  gives  2xx(ax  — xy)—{a2  + x2)X{ax—yx—xy)  ; 
, , . . x a"  + x2  x 

and  this  again  gives  - X . 

° & y 02  — X2  a_y 

Lastly,  this  value  of  — being  put  equal  the  former,  gives 

y -x 

a2  + x2 


368 


FLUXIONS. 


02  + X2 


a 2 + X2 


; and  hence  2x2  = a2  - xz . 

a—y 

the  absciss. 


a2  - x 2 ' a-y  2x 

or  3x2  — a2 , and  x = a 

Hence  also,  from  the  original  equation,  - • . , . 

ax  2 

y = = ~2  ~ 4a>  the  ordinate  of  the  point  of  in- 
flexion sought.  s 

Exam.  2.  To  find  the  point  of  inflexion  in  a curve  defined 
by  the  equation  ay  = a ax  + x2 . 

Exam.  3.  To  find  the  point  of  inflexion  in  a curve  defined 
by  the  equation  ay2  = a2x  + x3. 

Exam.  4.  To  find  the  point  of  inflexion 
in  the  Conchoid  of  Nicomedes,  which  is 
generated  or  constructed  in  this  manner  : 

From  a fixed  point  p,  which  is  called  the 
pole  of  the  conchoid,  draw  any  number  of 
right  lines  pa,  pb,  pc,  pe,  &.c.  cutting  the 
given  line  fd  in  the  points  f,  g,  h,  i,  &c.  : 
then  make  the  distances,  fa,  gb,  hc,  ie,  &c.  equal  to  each 
other,  and  equal  to  a given  line  ; then  the  curve  line  abce  &c. 
will  be  the  conchoid  ; a curve  so  called  by  its  inventor  Njco- 
medes. 


TO  FIND  THE  RADIUS  OF  CURVATURE  OF  CURVES 

108.  The  Curvature  of  a Circle  is  constant,  or  the  same 
in  every  point  of  it,  and  its  radius  is  the  radius  of  curvature. 
But  the  case  is  different  in  other  curves,  every  one  of  which 
has  its  curvature  continually  varying,  either  increasing  or 
decreasing,  and  every  point  having  a degree  of  curvature  pe- 
culiar to  itself ; and  the  radius  of  a circle  which  has  the  same 
curvature  with  the  curve  at  any  given  point,  is  the  radius  ef 
curvature  at  that  point ; which  radius  it  is  the  business  of  this 
chapter  to  find. 

109.  Let  AEe  be  any  curve,  con- 
cave towards  its  axis  ad  ; draw  an 
ordinate  de  to  the  point  e,  where 
the  curvature  is  to  be  found  : and 
suppose  ec  perpendicular  to  the 
curve,  and  equal  to  the  radius  of 
curvature  sought,  or  equal  to  the 
radius  of  a circle  having  the  same 
curvature  there,  and  with  that  ra- 
dius describe  the  said  equally 


FLUXIONS. 


369 


curved  circle  BEe  ; lastly,  draw  e d parallel  to  ad,  and  de  pa- 
rallel and  indefinitely  near  to  de  : thereby  making  e d the 
fluxion  or  increment  of  the  absciss  ad,  also  de  the  fluxion  of 
the  ordinate  de,  and  Ee  that  of  the  curve  ae.  Then  put 
x = ad,  y — de,  z = ae,  and  r — ce  the  radius  of  curva- 
ture ; then  is  e d = x,  de  = y , and  e e = z . 

Now,  by  sim.  triangles,  the  three  lines  e d,  de , ~e, 

or  x,  y , z , 

are  respectively  as  the  three  - - - ge,  gc,  ce  ; 

therefore - - - - gc  x :=  ge  . y ; 

and  the  flux,  of  this  eq.  is  gc  . x -f-  gc  . x — ge  . y gf,  . y\ 
or,  because  gc  = — bg  , it  is  gc  . x — bg  . x — ge  . y +ge  . y . 

But  since  the  two  curves  ae  and  be  have  the  same  cur- 
vature at  the  point  e,  their  abscisses  and  ordinates  have  the 
same  fluxions  at  that  point,  that  is,  Ed,  or  x is  the  fluxion 
both  of  ad  and  bg,  and  de  or  y is  the  fluxion  both  of  de 
and  ge.  In  the  equation  above  therefore  substitute  x for 
bg,  and  y for  ce,  and  it  becomes 

cc.r — xx  = gf y ~\~yy, 
or  GCx GFy  — xZ  ~\~y  2 — z2* 

Now  multiply  the  three  terms  of  this  equation  respectively, 
by-  these  three  quantities,  which  are  all  equal, 

and  it  becomes  - - - y'i — xy=  — , or  — ; 

• 3 

and  hence  is  found  r = — „ ~ t-g — , for  the  general  value  of 
y'x  — xy 

the  radius  of  curvature,  for  all  curves  whatever,  in  terms  of 
the  fluxions  of  the  absciss  and  ordinate. 

110.  Further,  as  in  any  case  either  x or  y maybe  supposed 
o flow  equably,  that  is,  either  x or  y constant  quantities,  or 
c or  y equal  to  nothing,  it  follows  that,  by  this  supposition, 
iither  of  the  terms  in  the  denominator,  of  the  value  of  r , 
nay  be  made  to  vanish.  Thus,  when  x is  supposed  constant 
i-  being  then  = 0,  the  value  of  r is  barely 

z3  . z3 

— rz-  : or  r is  = — when  y is  constant. 

-*!/  yi 

I 

EXAMPLES. 

i 

Exam.  1.  To  find  the  radius  of  curvature  to  any  point 
Vol.  II.  48  ' of 


37U 


INVOLUTES  AND  EVOLUTES. 


of  a parabola,  whose  equation  is  ax  = y2 , its  vertex  being  a 
and  axis  ad. 


Now,  the  equation  to  the  curve  being  ax  = y2 , the  fluxion 
of  it  is  ax  — lyy  ; and  the  fluxion  of  this  again  is  a^  = 2y  2 . 
supposing  y constant  ; hence  then  r or 


(xs  + y-y 


or 


■ is  ■ 


(a2  -f*  4y2)2  (a  -f*  4x)2 
or , 


yx  yx  2a2  2^/a 

for  the  general  value  of  the  radius  of  curvature  at  any  point 
e,  the  ordinate  to  which  cuts  off  the  absciss  ad  = x. 

Hence,  when  the  absciss  x is  nothing,  the  last  expression 
becomes  barely  fa  — r,  for  the  radius  of  curvature  at  the  ’ 
vertex  of  the  parabola  ; that  is,  the  diameter  of  the  circle  of 
curvature  at  the  vertex  of,  a parabola,  is  equal  to  a,  the  pa*  x 
rameter  of  the  axis. 


Exam.  2.  To  find  the  radius  of  curvature  of  an  ellip3e: 

— » . — , M A m A A . A ~ a — . M D . . O A A.  aa  a.  9 


whose  equation  is  a3y2  = c2  . ax  — x2 


Ans  a — /a2c2  +4(a2— c2)  X (ax-ga)  j | 
' ^ 2a* c ' ’ * 


Exam.  3.  To  find  the  radius  of  curvature  of  an  hyper- 


bola, whose  equation  is  a 2y2  = c2  . ax~\-x2. 


Exam.  4.  To  find  the  radius  of  curvature  of  the  cycloid. 
Ans.  r = 2^/da—ax,  where  x is  the  absciss,  and 
a the  diameter  of  the  generating  circle. 


OF  INVOLUTE  AND  EVOLUTE  CURVES. 


111.  An  Evolute  is  any  curve  supposed  to  be  evolved  or 
opened,  which  having  a thread  wrapped  close  about  it,  fasten- 
ed at  one  end,  and  beginning  to  evolve  or  unwind  the  thread 
from  the  other  end,  keeping  always  tight  stretched  the  part 
which  is  evolved  or  wound  off : then  this  end  of  the  thread 
will  describe  another  curve,  called  the  Involute.  Or,  the 
same  involute  is  described  in  the  contrary  way  by  wrapping 
the  thread  ahout  the  curve  of  the  evolute,  keeping  it  at  the 
same  time  always  stretched 

112.  Thus 


FLUXIONS. 


371 


112.  Thus,  if  efgh  be  any  curve, 
and  ae  be  either  a part  of  the  curve, 
or  a right  line  : then  if  a thread  be 
fixed  to  the  curve  at  h,  and  be 
wound  or  plied  close  to  the  curve, 

& c.  from  h to  a,  keeping  the  thread 
always  stretched  tight  ; the  other 
end  of  the  thread  will  describe  a 
certain  curve  abcd,  called  an  Invo- 
lute ; the  first  curve  efgu  being  its 
evolute.  Or,  if  the  thread,  fixed 
at  h.  he  unwound  from  the  curve,  beginning  at  a,  and 
it  always  tight,  it  will  describe  the  same  involute  abcd. 

113.  If  ae,  df,  cg,  dh,  &c.  be  any  positions,  of  the 
thread,  in  evolving  unwinding ; it  follows,  that  these 
parts  of  the  thread  are  always  the  radii  of  curvature,  at  the 
corresponding  points,  a,  b,  c,  d ; and  also  equal  to  the  cor- 
responding lengths  ae,  aef,  aefg,  aefgh,  of  the  evolute 
that  is, 

ae  = ae  is  the  radius  of  curvature  to  the  point  a, 

bf  = af  is  the  radius  of  curvature  to  the  point  b, 

cg  = ag  is  the  radius  of  curvature  to  the  point  c, 

dh  = ah  is  the  radius  of  curvature  to  the  point  d. 


114.  It  also  follows,  from  the  premises,  that  any  radius  of 
curvature,  bf,  is  perpendicular  to  the  involute  at  the  point  e, 
and  is  a tangent  to  the  evolute  curve  at  the  point  f.  Also, 
that  the  evolute  is  the  locus  of  the  centre  of  curvature  of  the 
involute  curve. 


115.  Hence,  and  from  art  109,  it 
will  be  easy  to  find  one  of  these 
curves,  when  the  other  is  given. 

To  this  purpose,  put 

x = ad,  the  absciss  of  the  involute, 
y — db,  an  ordinate  to  the  same, 
z = ab,  the  involute  curve, 
r — bc,  the  radius  of  curvature, 
v = ef,  the  absciss  of  the  evolute,  ec, 
u = fc,  the  ordinate  of  the  same,  and 
a,  — ae,  a certain  given  line. 


372 


INVOLUTES  AND  EVOLUTES. 


Then,  by  the  nature  of  the  radius  of  curvature,  it  is 

9 

r = — — ...  = bc  = ae  + ec  ; also,  by  sim.  triangles. 

yx  — xy 


Z : X : : r : gb 
2 ■ y : : r : gc 


y x ~ x y 
y i 


y * — Jry 
x 'z.  ‘ 


Hence  ef—gb — db  = — — y = v. 

V X — xy  3 


and  fc  = ad  — ae  -f-  gc  = x — a + ~ — u ; 

yx—xy 

which  are  the  values  of  the  absciss  and  ordinate  of  the 
evolute  curve  ec  : from  which  therefore  these  may  be  found, 
when  the  involute  is  given. 

On  the  contrary,  if  v and  «,  or  the  evolute,  be  given 
then,  putting  the  given  curve  ec  = s,  since  cb  = ae  + ec, 
or  r — a + s,  this  gives  r the  radius  of  curvature.  Also,  by 
similar  triangles,  there  arise  these  proportions,  viz. 


o • Tv  . 

5 * V • • T : — r-= : — v — GB, 

* 


J - ..  ru‘  ■ 

SDCl  5 ! u l l T l — 7 — U — GC  J 

3 3 

theref.  ad  = ae  + fc  — gc  = a + « — ^4—  u = xi 

3 

and  db  = gb  — gd  = gb  — ef  = ^4^  v — v — y ; 


which  are  the  absciss  and  ordinate  of  the  involute  curve,  and 
which  may  therefore  be  found,  when  the  evolute  is  given. 
Where  it  may  be  noted,  that  s2  = v2  + i\2,  and  z2  = x2  + y2  • 
Also,  either  of  the  quantities  x , y,  may  be  supposed  to  flow 
equably,  in  which  case  the  respective  second  fluxion,  x or  y, 
will  be  nothing,  and  the  corresponding  term  in  the  denomi- 
nator yx  — xy  will  vanish,  leaving  only  the  other  term  in  it ; 
which  will  have  the  effect  of  rendering  the  whole  operation 
simpler. 


116.  EXAMPLES. 


Exam.  i.  To  determine  the  nature  of  the  curve  by  whose 
evolution  the  common  parabola  ab  is  described. 


Here 


FLUXIONS. 


373 


Here  the  "equation  of  the  given  involute  ab,  is  cx  — y2 
where  c is  the  parameter  of  the  axis  ad.  Hence  then 

c — x~  c 

y — y cx,  and  y -\x-Ji — , also  y = - ^/-  by  making 

x 4x  x 

x constant.  Consequently  the  general  values  of  v and  u,  or 
of  the  absciss  and  ordinate,  ef  and  fc,  above  given,  become, 
in  that  case, 

ef  = v = — y = ■ y — 4x  — ; and 

— y — y c 

fc=u— x~a-i yAx.  — 3x  + \c  — a. 

— xy 

But  the  value  of  the  quantity  a or  ae,  by  exam.  1 to 
art-  75,  was  found  to  be  \c  ; consequently  the  last  quantity, 
fc  or  u is  barely  = 3x. 

Hence  then,  comparing  the  values  of  v and  u,  there  is 
found  3v^/c  = 3uy/x,  or  27 cv2  — 16w3  ; which  is  the  equa- 
tion between  the  absciss  and  ordinate  of  the  evolute  curve  ec, 
showing  it  to  be  the  semicubical  parabola. 

Exam.  2.  To  determine  the  evolute  of  the  common  cy- 
cloid. Ans.  another  cycloid,  equal  to  the  former. 


TO  FIND  THE  CENTRE  OF  GRAVITY. 


117.  By  referring  to  prop.  42,  &c.  in  Mechanics,  it  is  seen 
what  are  the  principles  and  nature  of  the  Centre  of  Gravity- 
in  any  figure,  and  how  it  is  generally 
expressed.  It  there  appears,  that  if 
paq  be  a line,  or  plane,  drawn  through 
any  point,  as  suppose  the  vertex  of  any 
body,  or  figure,  abd,  and  if  - - - 

s denote  any  section  ef  of  the  figure, 
d = ag,  its  distance  below  pq,  and 
b = the  whole  body  or  figure  abd  ; 
then  the  distance  ac,  of  the  centre  of 


sum  of  all  the  ds 
- 


gravity  below  pq,  is  universally  denoted  by 

whether  abd  be  a line,  or  a plane  surface,  or  a curve  super- 
ficies, or  a solid. 

> ■'  But 


■M 


374 


CENTRE  OF  GRAVITY. 


But  the  sum  of  all  the  ds , is  the  same  as  the  fluent  of  db, 
and  b is  the  same  as  the  fluent  of  b ; therefore  the  general 
expression  for  the  distance  of  the  centre  of  gravity,  is  ac  = 
fluent  of  ib  fluent  zb  . 

fluent  of  i ~ 1) ’ PuttlD§  x — “ the  variable  distance 

ag.  Which  will  divide  into  the  following  four  cases, 

118.  Case  1.  When  ae  is  some  line,  as  a curve  suppose. 

In  this  case  6 is  = z or  y/x2  -f  i/5,  the  fluxion  of  the  curve 

, , ,,  r fluent  of  x'x  fluent  of  m a2  4-  b2 

z z 

is  the  distance  of  the  centre  of  gravity  in  a curve. 

119.  Case  2.  When  the  figure  abd  is  a plane;  then 

b = yx  ; therefore  .ite  general  expression  becomes  ac  — 
fluent  of  yx'x  » ..  , ~ , , 

fluent  o f~^r  *or  tbe  distance  °t  the  centre  of  gravity  in  a 
plane. 


120.  Case  3.  When  the  figure  is  the  superficies  of  a body 
generated  by  the  rotation  of  a line  aeb,  about  th«  axis  ah. 
Then,  putting  c ==  3-J4159  &c.  2 cy  will  denote  the  circum- 
ference of  the  generating  circle,  and  2 cyi  the  fluxion  of  the 


surface  ; therefore  ac 


fluent  of  2 cyXz  fluent  of  1 x~ 


will 


fluent  of  2 cyi  fluent  of  yi 

be  the  distance  of  the  centre  of  gravity  for  a surface  generat- 
ed by  the  rotation  of  a curve  line  z. 


121.  Case  4.  When  the  figure  is  a solid  generated  by  the 
rotation  of  a plane  abh.  about  the  axis  ah. 

Then,  putting  c — 3-14159  &c.  it  is  cy2  =»  the  area  of 
the  circle  whose  radius  is  y,  and  cy2x  = b,  the  fluxion  of  the 

solid  ; therefore 

fluent  of  x i fluent  of  cysix  fluent  of  y2xi  . 

fluent  of  b~  fluent  of  cy2  i fluent  of  y2x 

the  distance  of  the  centre  of  gravity  below  the  vertex  in  a 
solid. 


122.  EXAMPLES. 


Exam.  1.  Let  the  figure  proposed  be  the  isosceles  triangle 

ABD. 

It  is  evident  that  the  centre  of  gravity  c,  will  be  some- 
where 


FLUXIONS. 


375 


where  in  the  perpendicular  ah.  Now,  if  a 
denote  ah,  c — bd,  x — ag,  and  y — ef 
any  line  parallel  to  the  base  bd  : then  as 
coc 

a : c : : x : y — — ; therefore,  by  the  2d 
fluent  yx'x  fluent  x^x  -g-x3 

Case,  ac  = — — = 

fluent  yic  fluent  Xx  $x2 

==  | x ==  | ah,  when  x becomes  = ah  : consequently  ch  = 

i AH. 

In  like  manner,  the  centre  of  gravity  of  any  othe^  Mane 
triangle,  will  be  found  to  be  at  i of  the  altitude  of  the  trian- 
gle ; the  same  as  it  was  found  in  prop  43,  Mechanics. 

Exam.  2.  In  a.  pafebola  ; the  distance  from  the  vertex  is 
%x,  or  | of  the  axis. 

Exam.  3.  In  a circular  arc  ; the  distance  from  the  centre 

of  the  circle,  is  — ; where  a denotes  the  arc,  c its  chord,  and 
a 

r the  radius. 

Exam.  4.  In  a circular  sector  ; the  distance  from  the  centre 
of  the  circle,  is  — : Where  a,  c,  r,  are  the  same  as  in  exam.  3« 
Exam.  5.  In  a circular  segment ; the  distance  from  the 

C 3 

centre  of  the  circle  is  — ; where  c is  the  chord,  and  a the 
1 2a 

area,  of  the  segment. 

Exam.  6.  In  a cone,  or  any  other  pyramid  ; the  distance 
from  the  vertex  is  f x,  or  | of  the  altitude. 

Exam.  7.  In  the  semisphere,  or  semispheriod  ; the  distance 
from  the  centre  is  fr,  or  f of  the  radius  : and  the  distance 
from  the  vertex  | of  the  radius. 

i 

Exam.  8.  In  the  parabolic  conoid  ; the  distance  from  the 
base  is  i#,  or  i of  the  axis.  And  the  distance  from  the  ver- 
tex f of  the  axis. 

Exam.  9.  In  the  segment  of  a sphere,  or  of  a spheriod  ; 

2j7— ^ . . 

■ the  distance  from  the  base  is  — — x ; where  x is  the  height 

6a-  4x  ’ 6 

of  the  segment,  and  a the  whole  axis,  or  diameter  of  the 
sphere. 

Exam.  10.  In  the  hyperbolic  conoid  ; the  distance  from 

K~}  (l  3C 

the  base  is  — ; — - x ; where  x is  the  height  of  the  conoid. 
6a+4x  ’ b 

and  « the  whole  axis  or  diameter. 


PRACTICAL 


[ 376  ] 


123.  PRACTICAL  QUESTIONS. 


QUESTION  L 


A large  vessel,  of  10  feet,  or  any  other  given  depth,  and 
of  any  shape,  being  kept  constantly  full  of  water,  by  means 
of  a supplying  cock,  at  the  top  ; it  is  proposed  to  assign  the 
place  where  a small  hole  must  be  made  in  the  side  of  it,  so 
that  the  water  may  spout  through  it  to  the  greatest  distance 
on  the  plane  of  the  base. 

Let  ab  denote  the  height  or  side  of 
the  vessel  ; d the  required  hole  in  the 
side,  from  which  the  water  spouts,  in 
the  parabolic  curve  dg,  to  the  greatest 
distance  bg,  on  the  horizontal  plane. 

By  the  scholium  to  prop.  68,  Hy- 
draulics, the  distance  bg  is  always  equal 
to  2 y/  ad  . db,  which  is  equal  to 

2 y/x  (x  — a)  or  2 y/  ax—x2,  if  a be  put  to  denote  the  whole 
height  ab  of  the  vessel,  and  x — ad,  the  depth  of  the  hole. 
Hence  2 y/  ax  — x2 , or  ax  — x2 , must  be  a maximum.  In 
fluxions,  a'x  — 2xx  = 0,  or  a — 2x  — 0,  and  2x  = a,  or 
x — \a.  So  that  the  hole  d must  be  in  the  middle  between 
the  top  and  bottom  ; the  same  as  before  found  at  the  end  of 
the  scholium  above  quoted. 


A 

D 

n 

124.  QUESTION  H. 

If  the  same  vessel,  as  in  Quest.  1,  stand  on  high,  with  its 
bottom  a given  height  above  a horizontal  plane  below  ; it  is 
proposed  to  determine  where  the  small  hole  must  be  made  so 
as  to  spout  farthest  on  the  said  plane. 


Let  the  annexed  figure  represent  the 
vessel  as  before,  and  6g  the  greatest  dis- 
tance spouted  by  the  fluid,  dg,  on  the 
plane  6g. 

Here,  as  before,  6g  = 2 ^/ad  . d6 
— 2 y/  x{c  — x')  = 2 y/  cx—x 2,  by 
putting  a b = c,  and  ad  — x.  So  that 
2 y/  cx—x 2 or  cx—x2  must  be  a max- 
imum. And  hence,  like  as  in  the  former  question,  - - - 

x = jc  — 4-a b.  So  that  the  hole  d must  be  made  in  the 

middle 


PRACTICAL  QUESTIONS. 


37? 


middle  between  the  top  of  the  vessel,  and  the  given  plane 
that  the  water  may  spout  farthest. 

125.  QUESTION  IlL 

But  if  the  same  vessel,  as  before,  stand  on  the  top  of  an 
inclined  plane,  making  a given  angle,  as  suppose  of  30  de- 
grees, with  the  horizon  ; it  is  proposed  to  determine  the 
place  of  the  small  hole,  so  as  the  water  may  spout  the  farthest 
on  the  said  inclined  plane. 

Here  again  (d  being  the  place  of  the 
hole,  and  bg  the  given  inclined  plane), 
ba  ==  2 ^/ad  d6  = 2 x{a  — x ± z ), 
putting  z — b b,  and,  as  before,  a = ab, 
and  x — ad.  Then  6g  must  still  be  a 
maximum,  as  also  b6,  being  in  a given 
ratio  to  the  maximum  bg,  oh  account 
of  the  given  angle  b.  Therefore  ax  — 

x2  ± xz,  as  well  as  z , is  a maximum.  Hence,  by  art.  54  of 
the  Fluxions,  ajc  — ~xx  ± zx  = 0,  or  a — 'ix  ±0=0; 
conseq.  ± z — 2 x — a;  and  hence  6g  = 2 ^/x{a  — x±z~) 
becomes  barely  2jc.  But  as  the  given  angle  gb 'o  is  = 30°, 
the  sine  of  which  is  i ; therefore  bg  = 2b6  or  2 z,  and  6g2  = 
bg2  — b62  = 3 z2  — 3 (2x?  — o)2 , or  ba  — ± (2x  — a)  3. 

Putting  now  these  two  values  of  ba  equal  to  each  other, 
gives  the  equation  2x  = ± (2r  — a)  3 , from  which  is  found 

x = ^ J a,  the  value  of  ad  required. 

n/3±1  4 ^ 

Note.  In  the  Select  Exercises,  page  252,  this  answer  is 


brought  out 


10 


a,  by  taking  the  velocity  proportional  to 


the  root  of  half  the  altitude  only. 


126.  QUESTION  IV. 


It  is  required  to  determine  the  size  of  a ball,  which,  being 
let  fall  into  a conical  glass  full  of  water,  shall  expel  the  most 
water  possible  from  the  glass  ; its  depth  being  6,  and  diame- 
ter 5 inches. 

Let  abc  represent  the  cone  of  the 
glass,  and  dhe  the  ball,  touching  the 
sides  in  the  points  d and  e,  the  centre 
of  the  ball  being  at  some  points  f in 
the  axis  gc  of  the  cone. 


Vor,,  If, 


49 


378  PRACTICAL  EXERCISES  ON  FORCEfe. 


Put  ag  ==  gb  = 2i  = a, 
cc  = 6 = b, 

AC  = AG2  + GC2  = = C, 

ad  = fe  = fh  = x the  radius  of  the  ball. 

The  two  triangles  acg  and  dcf  are  equiangular  ; there!'. 

ag  : ac  : : df  : fc,  that  is,  a : c : : x : — = fc  : hence 

’ a 


gf  = gc— fc  = b — — , and  gh  = gf  fh  = b + x — — 5 
a a 

the  height  of  the  segment  immersed  in  the  water  Then  (by 
rule  1 for  the  spherical  segment,  p.  427  vol.  1.),  the  content 
of  the  said  immersed  segment  will  be  (Gdf  — 2gh)  X gh2 

X ’5236  = (2 x-b  + — ) X (x  + b — —)2  X 1-0472,] 

v a ' v a ’ 


which  must  be  a maximum  by  the  question  ; the  fluxion  of 
this  made  = 0,  and  divided  by  2^  and  the  common  factors, 

gives X (b — f x — b)  X X 2 = 0 ; 

3 a v a ' ' a J a 


this  reduced  gives  x = r — - — — = 2ii,  the  ra- 

a (r_a)  x {.c  + 2.7)  9-’ 

dius  of  the  ball.  Consequently  its  diameter  is  4^|  inches,  as 
required. 


PRACTICAL  EXERCISES  CONCERNING  FORCES  , 
WITH  THE  RELATION  BETWEEN  THEM  AND 
THE  TIME,  VELOCITY,  AND  SPACE  DESCRIBED. 


Before  entering  on  the  following  problems,  it  will  be  con- 
venient here,  to  lay  down  a synopsis  of  the  theorems  which 
express  the  several  relations  between  any  forces,  and  their 
corresponding  times,  velocities,  and  spaces,  described  ; which 
are  all  comprehe  nded  in  the  following  12  theorems,  as  collect- 
ed from  the  principles  in  the  foregoing  parts  of  this  work. 

Let  /,  f,  be  any  two  constant  accelerative  forces,  acting  on 
any  body,  during  the  respective  times  t,  t,  at  the  end  of  which 
are  generated  the  velocities  v,  v,  and  described  the  spaces  s, 
s.  Then,  because  the  spaces  are  as  the  times  and  velocities 
conjointly,  and  the  velocities  as  the  forces  and  times  ; we 
shall  have, 

1.  In 


PRACTICAL  EXERCISES  ON  FORCES. 


37 


1.  In  Constant  Forces. 


1. 

tv 

_ t*f_ 

V2  F 

S 

TV 

T2  F 

2. 

V 

= J-L- 

ST 

—y/~- 

FS 

V 

FT 

St 

3. 

t 

TV 

SV 

. F? 

T 

~ Tv 

Sv 

4. 

/ 

TV 

T 2S 

V2  S 

— 

ZZ1  

— • -a 

F 

tv 

«2S 

V2f 

And  if  one  of  the  forces,  as 

f,  be  the 

force  of  gravity 

the  surface  of 

the 

earth,  and 

be  called 

1,  and  its  time 

be  = 1''  ; then  it  is  known  by  experiment  that  the  corres- 
ponding space  s is  = 16TV  feet,  and  consequently  its  velo- 
city v = 2s  ==  32|,  which  call  2 g,  namely,  g = 16-jL-  feet, 
or  193  inches.  Then  the  above  four  theorems,  in  this  case, 
become  as  here  below  : 


5. 

5 = 

¥v 

6. 

V = 

1 to 

t 

2s 

7. 

t = 

V 

8. 

/ = 

V 

2gi 

gfl2 

II 

2 gfi 

=V4S/*- 

V 

V 

t»2 

g{2 

' 

And  from  these  are  deduced  the  following  four  theorems, 
for  variable  forces,  viz. 


II.  In  Variable  Forces. 


9. 

5 

= VC 

Vv' 

~ ~2gf’ 

10. 

V 

= °~gfi 

— . 2g-/? 

V 

11. 

t 

S 

V 

V 

~~  2gP 

12. 

f 

e rj? 

1! 

II 

"sja' 

In 


380 


PRACTICAL  EXERCISES  ON  FORCES 


In  these  last  four  theorems,  the  force  /,  though  variable, 
is  supposed  to  be  constant  for  the  indefinite!}'  small  time  >, . 
and  the}  are  to  be  used  in  all  cases  of  variable  forces,  as  the 
former  ones  in  constant  forces  ; namel}  from  the  circum- 
stances of  the  problem  under  consideration  an  expression  is 
deduced  for  the  value  of  the  force  f,  which  being  substituted 
in  one  of  these  theorems,  that  may  be  proper  to  the  case  in 
hand  ; the  equation  thence  resulting  will  determine  the  > 
corresponding  values  of  the  other  quantities,  required  in  the  ; 
problem. 

When  a motive  force  happens  to  be  concerned  in  the 
question,  it  ipa}  be  proper  to  observe,  that  the  motive  force 
m,  of  a body  is  equal  to fq,  the  product  of  the  accelerative 
force,  and  the  quantity  of  matter  in  it  q ; and  the  relation 
between  these  three  quantities  being  universally  expressed 
by  this  equation  m — qj,  it  follows  that,  by  means  of  it,  any  ■ 
one  of  the  three  may  be  expelled  out  of  the  calculation,  or  ! 
else  brought  into  it. 

Also,  the  momentum,  or  quantity  of  motion  in  a moving  ;; 
body,  is  qv , the  product  of  the  velocity  and  matter. 

It  is  also  to  be  observed,  that  the  theorems  equally  hold 
good  for  the  destruction  of  motion  and  velocity,  by  means  of 
retarding  forces,  as  for  the  generation  of  the  same,  by  means  1 
of  accelerating  forces. 

Ami  to  the  following  problems,  which  are  all  resolved  by  ! 
the  application  of  these  theorems,  it  has  been  thought  proper 
to  subjoin  their  solutions,  for  the  better  information  and  con- 
venience of  the  student. 


PROBLEM  I. 

To  determine  the  time  and  velocity  of  a body  descending,  by  the 
force  of  gravity,  down  an  inclined  plane  ; the  length  of  the 
plane  being  20  feet,  and  its  height  1 foot. 


Here,  by  Mechanics,,  the  force  of  gravity  being  to  the 
force  down  the  plane,  as  the  length  of  the  plane  is  to  its 
height,  therefore  as  20  : 1 : : 1 (the  force  of  gravity)  : = 

f the  force  on  the  plane. 

Therefore,  by  theor.  6,  v cr ^/4gfs  is  4 X 16TV  X jfe  X 
20  = 4 X 1£tV  = 2 X 4?Jg  or  feet  nearly,  the  last 

velocity  per  second.  And, 

, s . 20  400  20 

By  theor.  7,  t or  ^/-.is^t rrr~r 


«/—  .is  v/  - — 

gj  i«§x5v 
seconds,  the  time  of  descending. 


16, 


PROBLEM 


PRACTICAL  EXERCISES  ON  FORCES. 


381 


PROBLEM  n. 


If  a cannon  ball  be  fired  with  a velocity  of  1000  feet  per  second 
up  a smooth  inclined  plane,  which  rises  1 foot  in  20:  it  is 
proposed  to  assign  the  length  which  it  will  ascend  up  the  plane, 
before  it  stops  and  begins  to  return  down  again,  and  the  time 
of  its  ascent. 

Here/  =YV  as  before. 

v2'  10002  60000000 

Then,  b,  theor.  5,  . = —f  = = — jsj- 

= 310880i||  feet,  or  nearly  59  miles,  the  distance  moved. 

. , , / „ v 1000  120000 

And,  by  theor,  7,  < = -f  = k ^ — r 


193 


i621"  J/i 


10'  21"  j-Ai,  the  time  of  ascent. 


PROBLEM  III. 


If  a ball  be  projected  up  a smooth  inclined  plane,  which  rises  1 
foot  in  10.  and  ascend  100  feet  before  it  stop  : required  the 
time  of  ascent,  and  the  velocity  of  projection. 


First,  by  theor.  6,  v — 4 gfs  -/4X  1 6/^  X TV  X 

100  = 8t'¥  10  = 25-36408  feet  per  second,  the  velocity. 

And,  by  theor.  7 ,t  = v'  lt;_.  iX  S 10 

bj  Ut12  A 10  '96 

==  Ls/  10  = 7-88516  seconds,  the  time  in  motion. 


PROBLEM  IV. 

If  a ball  be  observed  to  ascend  up  a smooth  inclined  plane  : 
100  feet  in  10  seconds,  before  it  stop,  to  return  back  again 
required  the  velocity  of  projection,  and  the  angle  of  the 
plane's  inclination. 

First,  by  theor.  6,  v = — = 222  = 20  feet  per  second, 
the  velocity. 

s 100  12 

And,  by  theor.  8 ,/=  — = — — That 

J J gc-  167V  X 100  ir- 

is, the  length  of  the  plane  is  to  its  height,  as  193  to  12. 

Therefore  193  : 12  : : 100  : 6-2176  the  height  of  the 
plane,  or  the  sine  of  elevation  to  radius  100,  which  answers  to 
3°  34',  the  angle  of  elevation  of  the  plane. 


PROBLEM 


382 


PRACTICAL  EXERCISES  ON  FORCES 


PROBLEM  V. 

By  a mean  of  several  experiments,  I have  found,  that  a east 
iron  ball,  of  2 inches  diameter , fired  perpendicularly  into  the 
face  or  end  of  a block  of  elm  -wood,  or  in  the  direction  of  the 
fibres,  with  a velocity  of  IbOOfeet  per  second,  penetrated  15 
inches  deep  into  its  substance.  It  is  proposed  then  to  deter- 
mine the  time  of  the  penetration,  and  the  resisting  force  of 
the  wood,  as  compared  to  the  force  of  gravity,  supposing 
that  force  to  be  a constant  quantity. 


2 s 


2X  13 
1500X  12 

15002 


= — part  of  a se- 

692  r 


First,  by  theor.  7,  t = — = 

cond,  the  time  in  penetrating. 

, , , , „ . v2  15002  eiOOnoOO 

A-d, ,heor- s-  = 4 x I6J,XH  = irsnss 

= 32284.  That  is,  the  resisting  force  of  the  wood,  is  to 
the  force  of  gravity,  as  32284  to  1 . 

But  this  number  will  be  different,  according  to  the  dia- 
meter of  the  ball,  and  its  density  or  specific  gravity.  For, 

v2 

since/  is  as  — by  theor.  4,  the  density  and  size  of  the  ball 

remaining  the  same  ; if  the  density,  or  specific  gravity,  n, 
v&ry,  and  all  the  rest  be  constant,  it  is  evident  that/ will 

be  asn  ; and  therefore /as when  the  size  of  the  ball  only 

is  constant.  But  when  only  the  diameter  d varies,  all  the 
rest  being  constant,  the  force  of  the  blow  will  vary  as  d3  or 
as  the  magnitude  of  the  ball  ; and  the  resisting  surface,  or 

d 3 

force  of  resistance,  varies  as  d2  ; therefore/  is  as  — or  as  d 
only  when  all  the  rest  are  constant.  Consequently  / is  as 
•— — when  they  are  all  variable. 

F where  / denote  the 


/ dnv2 s - 

And  so  - = — and  - = - , 

F D»VJt  s DNv2y 


strength  or  firmness  of  the  substance  penetrated,  and  is  here 
supposed  to  be  the  same,  for  all  balls  and  velocities,  in  the 
same  substance,  which  is  either  accurately  or  nearly  so.  See 
page  581,  &c-  vol.  1,  of  my  Tracts. 

Hence,  taking  the  numbers  in  the  problem,  it  is 
= *»i=  ft  X7t  XJ500i=  44  X ^ 

s t!  39 

value  of  / for  elm  wood.  Where  the  specific  gravity  oi 

the 


PRACTICAL  EXERCISES  ON  FORCES. 


383 


the  ball  is  taken  7i,  which  is  a little  less  than  that  of  solid 
cast  iron,  as  it  ought,  on  account  of  the  air  bubble  which  is 
found  in  all  cast  balls. 

. 

PROBLEM  VI. 

■ | 

To  find  how  far  a 24lb  ball  of  cast  iron  will  penetrate  into  a 
block  of  sound  elm , when  fired  with  a velocity  of  1600  feet 
per  second. 


Here,  because  the  substance  is  the  same  as  in  the  last 
problem,  both  of  the  balls  and  wood  n = n,  and  f — f ; 


therefore  - = 


dv 


or  s 


dv2 


inches  nearly,  the  penetration  required 


2x15003 


= 41-3- 

'45 


PROBLEM  Vn. 

| 

It  was  found  by  Mr.  Robins,  (vol.  i.  p.  273,  of  his  works), 
that  an  18-pounder  ball,  fired  with  a velocity  of  1200  feet 
per  second,  penetrated  34  inches  into  sound  dry  oak.  It  is  re- 
quired then  to  ascertain  the  comparative  strength  or  firmness 
of  oak  and  elm. 


The  diameter  of  a 161b  ball  is  5 04  inches  = d.  Then, 
by  the  numbers  given  in  this  problem  for  oak,  and  in  prob.  5, 

for  elm,  we  have  

f __dv 2s  _ 2 X15002X34  _ 100X17  _ 1700  g 

f dv2s  5 04  X 12002  x 13  _ 5 04  x 16  X 1 3 ~~  1048  °r  * 
nearly. 

From  which  it  would  seem,  that  elm  timber  resists  more 
than  oak,  in  the  ratio  of  about  8 to  5 ; which  is  not  probable 
as  oak  is  a much  firmer  and  harder  wood.  But  it  is  to  be 
suspected  that  the  great  penetration  in  Mr.  R’s  experiment 
was  owing  to  the  splitting  of  his  timber  in  some  degree. 


PROBLEM  VIII. 

A 24-pounder  ball  being  fired  into  a bank  of  firm  earth,  with  a 
velocity  of  1300  feet  per  second,  penetrated  15  feet.  It  is 
required  then  to  ascertain  the  comparative  resistance  of  elm 
and  earth. 


Comparing  the  numbers  here  with  those  in  prob.  5,  it 

is 


I 


384 


PRACTICAL  EXERCISES  OX  FORCES. 


. / _ ch-2  s _ 2 x 15(T  2 X 15  X Ia  __  152  x 24  _ 
f Dv2i  b x ••  uu  ' X L.  133  Xfr  7 
Yt0t#  = 3J  nearly  =•=  6 | nearly.  That  is,  elm  timber  resists  | 
about  6|  times  more  than  earth. 


PROBLEM  IX. 

To  determine  how  far  a leaden  bullet,  of  | of  an  inch  diameter , I 
will  penetrate  dry  elm ; supposing  it  fired  with  a velocity  of  I 
1700  feet  per  second,  and  that  the  lead  does  not  change  its  I 
figure  by  the  stroke  against  the  wood. 


Here  d = f,  n = 11|,  n = 7L.  Then  by  the  numbers 

and  theorem  in  prob.  5,  it  is  s - 1 

DNV2„.  fxiljx  17002  X 13  173  x 13  63869  _ 

dnv 2 ~ Tx  7j  X 150^3  2 00  x 33  6600  I 

9|  inches  nearly,  the  depth  of  penetration. 

But  as  Mr  Robins  found  this  penetration,  by  experiment,  : 
to  be  only  5 inches  ; it  follows  either  that  his  timber  must  . 
have  resisted  about  twice  as  much  ; or  else,  which  is  much 
more  probable,  that  the  defect  in  his  penetration  arose  from 
the  change  of  figure  in  the  leaden  bail  he  used,  from  the 
blow  against  the  wood. 


PROBLEM  X. 

A one  pound  ball,  projected,  with  a velocity  of  1500  feet  per 
second,  having  been  found  to  penetrate  13  inches  deep  into 
dry  elm : It  is  required  to  ascertain  the  time  of  passing 
through  every  single  inch  of  the  13,  and  the  velocity  lost  at 
each  of  them  ; supposing  the  resistance  of  the  Wood  constant 
or  uniform. 


The  velocity  v being  1500  feet,  or  1500  X 12  = 18000 
inches,  and  velocities  and  times  being  as  the  roots  of  the 
spaces,  in  constant  retarding  forces,  as  well  as  in  accelerating 


2s 


ones,  and  t being  = — — 


26 


K 
P000 


— part  of 

692  * 


12  X -.100 

a second,  the  whole  time  of  passing  through  the  13  inches  ; 
therefore,  as 


VI? 


PRACTICAL  EXERCISES  ON  FORCES. 


385 


v/13  : yiS  — ^12  : : v : 


veloc.  lost 


Time  in  the 


| v'Ij-  v'IS 
v/13 

y/12-  v/ll 

I v/13 


v = 58‘9  : : t : 
•y  = 61'4  : : t : 


64-2  &c. 


I v/13 

,*67-5 

v/  13 

, = 71-4 


v/ 13 

^8-^7  = 76  0 


v/13 

^LT-^Lv  = 81-7 
v/13 

yS—^/5  v _ 88.g 


v'IS 

— " '/4  « = 98-2 


■✓13 

v/4-  v/3  _ , « . , 

7,-r V — 111'4 

V13 

132’2 

V,’2_v/1  V = 172-3 


v'IS 
Vl-v/0 

-is—7’- 41S'° 


v/ 13—  v/ 12  

v/13 

v/12-v/H 

v/Ts 

^/ll— v/10 
v/13 

v/lO-v/9 


v/13 

v/9-y/8 

v/13 

v/8-v/7 

v/13 

y/~—  y/6 

v/13 

y/6— y/5 
v/13 

y/5— y/4 
v- 13 

s/4- y/3 
v/13 

v/3—  v/2 
v/13- 
v/2-y/l 

V/13 

v'l-y'O 

v/13 


r = 


< = 


t = 


00005  1st  inch,, 
•00006  2d 
•00006  3d 
•00007  4th 
•00007  5th 
•00007  6th 
•00008  7th 
•00008  8th 
•00009  9th 
•00011  10th 
•00013  11th 
•00017  12th 
•00040  13th  ' 


Sum  1500’0 


Sum  ^ or  *00144  sec. 


Hence,  as  the  motion  lost  at  the  beginning  is  very  small  ; 
nd  consequently  the  motion  communicated  to  any  body,  as 
n inch  plank,  in  passing  through  it,  is  very  small  also  ; we 
an  conceive  how  such  a plank  may  be  shot  through,  when 
tanding  upright,  without  oversetting  it. 


tol.  II. 


50 


PROBLEM 


*'■*»<*» 


386 


PRACTICAL  EXERCISES  ON  FORCES 


PROBLEM  XL 


The  force  of  attraction,  above  the  earth,  being  inversely  as  the 
square  of  the  distance  from  the  centre  ; it  is  proposed  to  deter- 
mine the  time,  velocity,  and  other  circumstances,  attending  a 
heavy  body  falling  from  any  given  height  ; the  descent  at  the 
earth1  s surf  ace  being  16^  feet,  or  193  inches,  in  the  first  second 
of  time. 


Put 


t — cs  the  radius  of  the  earth, 
a = ca  the  dist.  fallen  from, 
x — cp  any  variable  distance, 
v — the  velocity  at  p, 
t = time  of  falling  there,  and 
— 167'7,  half  the  veloc.  or  force  at  s, 
= the  force  at  the  point  p. 


Then  we  have  the  three  following  equations,  viz. 
x-  : r-  : : 1 : f 1_  the  force  at  p,  when  the  force  of 

J X2 

gravity  is  considered  as  1 ; tv  =— x,  because  x decreases  ; and 


But 


The  fluents  of  the  last  equation  give  = — 
when  x = a,  the  velocity  v = 0 ; therefore,  by  correction. 

v,  = 4S^_V;  = 4 , x »_=£  . oc  „ = . (J£2  xq, 

a general  expression  for  the  velocity  at  any  point  p. 

When  t — r,  this  gives  v — y/  (4 gr  X - — -)  for  the 

greatest  velocity,  or  the  velocity  when  the  body  strikes  the 
earth. 


When  a is  very  great  in  respect  of  r,  the  last  velocity  be- 
comes (1  — — ) X y/  4 gr  very  nearly,  or  nearly  y/  4gr  only, 

which  is  accurately  the  greatest  velocity  by  falling  from  an 
infinite  height.  And  this,  when  r = 3965  miles,  is  6-9506 
miles  per  second.  Also,  the  velocity  acquired  in  falling  from 

the 


PRACTICAL  EXERCISES  ON  FORCES. 


'387 


the  distance  of  the  sun,  or  12000  diameters  of  the  earth,  in 
6-9505  miles  per  second.  And  the  velocity  acquired  in  falling 
from  the  distance  of  the  moon,  or  30  diameters,  is  6-8972 
miles  per  second. 


Again,  to  find  the  time  ; since 
— xk 


. — x a 

: Z ^ 4gr2 


ax—: 


t-o  = 

the 


= — x,  therefore 
correct  fluent  of 


which  gives  t — y/  - — - X (y/ax  — xx  -f-  arc  to  diameter  a 

and  vers,  a — x)  ; or  the  time  of  falling  to  any  point  r — 

h v/  lx  (ab  -f-  bp).  And  when  x = r,  this  becomes 
S 


2 r 


t = ix/~ 
2 V g 


AD  -f-DS 


SC 


for  the  whole  time  of  falling  to  the 

surface  at  s ; which  is  evidently  infinite  when  a or  ac  is  infi- 
nite, though  the  velocity  is  then  only  the  finite  quantity  x / , 
*gr 

When  the  height  above  the  earth’s  surface  is  given  = g ; 
because  r is  then  nearly  — a,  and  ad  nearly  = ds,  the  time  t 
for  the  distance  g will  be  nearly  - ------  -*  - 

X2ds=vA^  X y/  4 gr  = \'\  as  it  ought  lo  be. 

If  a body,  at  the  distance  of  the  moon  at  a,  fall  to  the 
earth’s  surface  at  s.  Then  r = 3965  miles,  a — 60 r,  and 
t = 416806''  = 4 da.  19  h.  46'  46",  which  is  the  time  of  fall- 
ing from  the  moon  to  the  earth. 

When  the  attracting  body  is  considered  as  a point  c ; the 

whole  time  of  descending  to  c will  be-- 

1 a -7854a  a 10a  "7854  as 

-V7-X  ABDC  = y/  - = -~y/a  = v/ — . 

2rv  g r v g 5lrv  r v g 

Hence,  the  times  employed  by  bodies,  in  falling  from 
quiescence  to  the  centre  of  attraction,  are  as  the  square  roots 
«f  the  cubes  of  the  heights  from  which  they  respectively  fall. 


PROBLEM  X1L 

Tile  force  of  attraction  below  the  earth’s  surface  being  directly 
as  the  distance  from  the  centre  ; it  is  proposed  to  determine 
the  circumstances  of  velocity,  time,  and  space  fallen  by  a heavy 
body  from  the  surface,  through  a perforation  made  straight  to 
the  centre  of  the  earth  : abstracting  from  the  effect  of  the 
earth's  rotation,  and  supposing  it  to  be  a homogeneous  sphere 
of  3965  miles  radius. 


Put 


388 


PRACTICAL  EXERCISES  ON  FORCES. 


Put  r — ac  the  radius  of  the  earth, 
x = cp  the  dist.  from  the  centre, 
v — the  velocity  at  p, 
t = the  time  there, 
g = 16tl,  half  the  force  at  a, 
f = the  force  at  p 
Then  ca  : cp  : : 1 and  the  three 

equations  are  rf  — x,  and  v'v  — — 2^2,  and  v't  — — x- 
lienee/  — — , and  v'v  the  correct  fluent  of 

^2  2 ^ nr 

which  gives  v = y/  (2 g X y=  pd  y/~—  =pdv/^- _,  the 

velocity  at  the  point  f ; where  pd  and  ce  are  perpendicular 
to  ca.  So  that  the  velocity  at  any  point  p,  is  as  the  perpen- 
dicular or  sine  pd  at  that  point. 

When  the  bofly  arrives  at  c,  then  v — ^/  2gr  — ^/2g  . ac 
= 25950  feet  or  4 9148  miles  per  second,  which  is  the  great- 
est velocity,  or  that  at  the  centre  c. 

Again,  for  the  time  ; 't  = — = +/  — X — -=-~ — : and  the 
* 2g  y/r*  —X2 

fluents  give  t =*=  X arc  to  cosine  ■—  = X arc 

ad.  So  that  the  time  of  descent  to  any  point  p,  is  as  the  cor- 
responding arc  ad 


When  p arrives  at  c,  the  above  becomes  t = - - - - 

a/ —■  quadrant  ae  ——*/ — = T5708  +/ — =12671  se- 

V 1 ACV  i g 

Conds  = 21 '7,fi,  for  the  time  of  falling  to  the  centre  c. 


The  time  of  falling  to  the  centre  is  the  same  quantity. 

T5708  -A,  from  whatever  point  in  the  radius  ac  the 

body  begins  to  move.  For,  let  n be  any  given  distance  from 
c at  which  the  motion  commences  : then  by  correction, 

^ nr  ■ 1 ~ i ^ y mmm 

v = x/  (—  .»i2—  x2),  and  hence  t = ^ — X . the 

r *g  V„2_j:2 

fluents  of  which  give  t = */  — - X arc  to  cosine  — ; which, 

2g  n 

when  x <=--  0,  gives  t = ~ X quadrant  = T5708  ^ 

for  the  time  cf  descent  to  the  centre,  c,  the  same  as  before. 

i.-.-..  As 


PRACTICAL  EXERCISES  ON  FORCES. 


389 


As  an  equal  force,  acting  in  contrary  directions,  generates 
or  destroys  an  equal  quantity  of  motion,  in  the  same  time  ; 
it  follows  that,  after  passing  the  centre,  the  body  will  just 
ascend  to  the  opposite  surface  at  b,  in  the  same  time  in 
which  it  fell  to  the  centre  from  a Then  from  b it  will 
return  again  in  the  same  manner,  through  c to  a : and  so 
oscillate  continually  between  a and  b,  the  velocity  being 
always  equal  at  equal  distances  from  c on  both  sides  ; and 
the  whole  time  of  a double  oscillation,  or  of  passing  from  a 
and  arriving  at  a again,  will  be  quadruple  the  time  of  passing 

over  the  radius  ac,  or  = 2 X 3T416  */  — = lh.  24'  29". 

V 2 g 


PROBLEM  XIII. 


To  find,  the  Time  of  a Pendulum  -vibrating  in  the  Arc  of  a 
Cycloid. 


S 


Let 

s be  the  point  of  suspension  ; 
sa,  the  length  of  pendulum  ; 
cab,  the  whole  cycloidal  arc ; 
aikd,  the  generating  circle, 
to  which  fee,  hig  are  per- 
pendiculars. 

sc,  sb  two  other  equal  se- 
micloids,  on  which  the 
thread  wrapping,  the  end 
a is  made  to  describe  the 
cycloid  bac. 

Now,  by  the  nature  of  the  cycloid,  ad  = ns  ; and  sa  = 
2ad  ~ sc  = sb  = sa  = ab.  Also,  if  at  any  point  a be 
drawn  the  tangent  gp  ; also  gq.  parallel  and  pq,  perpendicular 
to  ad.  Then  pg  is  parallel  to  the  chord  ai  by  the  nature  of 
the  curve.  And,  by  the  nature  of  forces,  the  force  of  gravity: 
force  in  direction  gp  : cp  : gq  : : ai  : ah  : : ad  : ai  ; in 
like  manner,  the  fgrce  of  gravity  : force  in  the  curve  at  e : : 
ad  : ak  ; that  is,  the  accelerative  force  in  the  curve,  is  every 
where  as  the  corresponding  chord  ai  or  ak  of  the  circle,  or  as 
the  arc  ag  or  ae  of  the  cycloid,  since  ag  is  always  = 2ai, 
by  the  nature  of  the  curve.  So  that  the  process  and  conclu- 
sions, for  the  velocity  and  time  of  describing  any  arc  in  this 
.case,  will  be  the  very  same  as  in  the  last  problem,  the  nature 
of  the  forces  being  the  same,  viz.  as  the  distance  to  be  passed 
^ver  to  the  lowest  point  a. 


From 


390 


PRACTICAL  EXERCISES  OX  FORCES. 


From  which  it  follows,  that  the  time  of  a semi-vibratioiv 
in  all  arcs,  ag,  ae,  &c.  is  the  same  constant  quantity 

1-5708  y/ 


2g 


AC  / 

1 5708  y/  — — 1'5708  y/  — ; and  the  time 

*S  *g 

of  a whole  vibration  from  b to  c,  or  from  c to  b,  is  3-1416—  ; 

~g 

where  l — as  = ab  is  the  length  of  the  pendulum,  g = 16r'T 
feet  or  193  inches,  and  3-1416  the  circumference  of  a circle 
whose  diameter  is  1. 


Since  the  time  of  a body’s  falling  by  gravity  through  JZ, 
or  half  the  length  of  the  pendulum,  by  the  nature  of  descents, 

is  -/— , which  being  in  proportion  to  3-1416  y/  — , as  1 is  to 

~g  2g  | 

3-1416  ; therefore  the  diameter  of  a circle,  is  to  its  circum-  \ 
ference,  as  the  time  of  falling  through  half  the  length  of  a 
pendulum,  is  to  the  time  of  one  vibration. 


If  the  time  of  the  whole  vibration  be  1 second,  this  equa- 

• a l "2(r  & 

tion  arises,  viz.  1 =3-1416./  — ; hence  Z=  ■■  ■-  = - _ ■ 

v 2 g ’ 3-14163  4-9348 

and  g = 3-14162  X f Z = 4 93481.  So  that  if  one  of  these,  I 


g or  Z,  be  given  by  experiment,  these  equations  will  give  i 
the  other.  When  g,  for  instance,  is  supposed  to  be  given  | 

_ g 


= 16  T'5  feet,  or  193  inches  ; then  is  Z 


= 39-11,  the  1 


4-9348 

length  of  a pendulum  to  vibrate  seconds.  Or  if  l = 39i, 
the  length  of  the  seconds  pendulum  for  the  latitude  of  Lon- 
don, by  experiment  ; then  is  g = 4-9348Z  = 193  07  inches 
— IfifiVo  feeL  or  nearly  16TV  feet,  for  the  space  descended 
by  gravity  in  the  first  second  of  time,  in  the  latitude  of  Lon- 
don ; also  agreeing  with  experiment. 


Hence  the  times  of  vibration  of  pendulums,  are  as  the 
square  roots  of  their  lengths  ; and  the  number  of  vibrations 
made  in  a given  time,  is  reciprocally  as  the  square  roots  of 
the  lengths.  And  hence  also,  the  length  of  a pendulum 
vibrating  n times  in  a minute,  or  60',  is  Z = 39i  X 
GO2  __  140850 
71  nn 

When  a pendulum  vibrates  in  a circular  arc  ; as  the  length 
of  the  string  is  constantly  the  same,  the  time  of  vibration 
will  be  longer  than  in  a cycloid  ; but  the  two  times  will  ap- 
proach nearer  together  as  the  circular  arc  is  smaller  ; so  that 

when 


PRACTICAL  EXERCISES  ON  FORCES. 


391 


when  it  is  very  small,  the  times  of  vibration  will  he  nearly 
equal  And  hence  it  happens  that  39f  inches  is  the  length 
of  a pendulum  vibrating  seconds,  in  the  very  small  arc  of  a 
circle. 

PROBLEM  XIV. 

- To  determine  the  Time  of  a Body  descending  dorm  the  Chord 
of  a Circle. 

Let  c he  the  centre  ; ab  the  vertical 
diamete,r  ; ap  any  chord,  down  which  a 
body  is  to  descend  from  p to  a ; and  pq 
perpendicular  to  ab. 

Now,  as  the  natural  force  of  gravity  in 
the  vertical  direction  ba,  is  to  the  force 
> urging  the  body  down  the  plane  pa,  as 
i the  length  of  the  plane  ap,  is  to  its  height 
aq  ; therefore  the  velocity  in  pa  and  qa, 
will  he  equal  at  all  equal  perpendicular 
distances  below  pq  ; and  consequently  the 
time  in  pa  : time  in  qa  : : pa  : qa  : : ba  : pa  ; but 

time  in  ba  : time  in  qa  : : ba  : y'  qa  : : ba  : pa  ; 

hence,  as  three  of  the  terms  in  each  proportion  are  the 

same,  the  fourth  terms  must  be  equal,  namely  the  time  in 

ba  = the  time  pa. 

And,  in  like  manner,  the.  time  in  bp  = the  time  in  ba. 
| So  that,  in  general,  the  times  of  descending  down  all  the 
? chords  ba,  bp,  bk,  bs,  &c.  or  pa,  ha,  sa,  &c.  are  all  equal, 
and  each  equal  to  the  time  of  falling  freely  through  the 
diameter  ; as  before  found  at  art.  131,  Mechanics.  Which 

I • 

time  is  y'  — , where  g = 16tl  feet,  and  r — the  radius  ac  : 

for  VS  ■ V2r  • 1''  : V 

PROBLEM  XV. 

! To  determine  the  Time  of  filling  the  Ditches  of  a JVork  with 
Water,  at  the  Top,  by  a Sluice  of  2 Feet  square  ; the  Head 
of  Water  above  the  Sluice  being  10  Feet,  and  the  Dimensions 
of  the  Ditch  being  20  Feet  wide  at  Bottom,  22  at  Top,  9 deep, 
and  1000  Feet  long. 

■B .....  h '■ 

Thf.  capacity  of  the  ditch  is  189000  cubic  feet. 

But  a/  g ■ a/  10  : : 2 g : 2^/lOgthe  velocity  of  the  water 
through  the  sluice,  the  area  of  which  is  4 square  feet  : 

i therefore 


302 


PRACTICAL  EXERCISES  ON  FORCES. 


therefore  8^/1  Og  is  the  quantity  per  second  running  through 

23625 

it ; and  consequently  8^/10 g : 189000  : : 1"  : — ---  = 1863" 
or  31'  3 ' nearly,  which  is  the  time  of  filling  the  ditch. 


PROBLEM  XVI. 


To  determine  the  Time  of  emptying  a Vessel  of  Water  by  a 
Sluice  in  the  Bottom  of  it,  or  in  the  Side  near  the  Bottom 
the  Height  of  the  -Aperture  being  very  small  in  respect  of  the 
Altitude  of  the  Fluid. 


Put  a = the  area  of  the  aperture  or  sluice  ; 

2 g = 32^  feet,  the  force  of  gravity  ; 
d = the  whole  depth  of  water  ; 

x = the  variable  altitude  of  the  surface  above  the 
aperture  ; 

a = the  area  of  the  surface  of  the  water. 


Then  y/  g : y/  x : : 2g  : 2 y/gx  the  velocity  with  which  the 
fluid  will  issue  at  the  sluice  ; and  hence  a : a : : 2 y/gx 

A 

the  velocity  with  which  the  surface  of  the  water  will  descend 
at  the  altitude  x,  or  the  space  it  would  descend  in  1 second 
with  the  velocity  there.  Now,  in  descending  the  space  x, 
the  velocity  may  be  considered  as  uniform  ; and  uniform  de- 
scents are  as  their  times  ; therefore  : f : : i" : — — — 

’ A 2 ay/gx 

the  time  of  descending  x space,  or  the  fluxion  of  the  time  of 
exhausting.  That  is,  t = ; which  is  made  negative, 

because  x is  a decreasing  quantity,  or  its  fluxion  negative. 


Now,  when  the  nature  or  figure  of  the  vessel  is  given,  the 
area  a will  be  given  in  terms  of  x ; which  value  of  a being 
substituted  into  this  fluxion  of  the  time,  the  fluent  of  the  result 
will  be  the  time  of  exhausting  sought. 


So  if,  for  example,  the  vessel  be  any  prism,  or  every 
where  of  the  same  breadth  ; then  a is  a constant  quantity, 

and  therefore  the  fluent  is  — - . But  when  x — d,  this 

a v g 


becomes 


fluent  is  t = — X 
a 


, and  should  be  0 ; therefore  the  correct 

■v/c  ~<>/--  for  the  time  of  the  surface  de- 
>/S  ,. 

scending 


PRACTICAL  EXERCISES  ON  FORCES^ 


393 


scending  till  the  depth  of  the  water  be  x.  And  when  x = 0, 
the  whole  time  of  exhausting  is  barely  -^/ 

Hence,  if  a be  = 10000  square  feet,  a = 1 square  foot, 
and  d = 10  feet ; the  time  is  7885|  seconds,  or  2h  1 1'  25"-i. 

Again,  if  the  vessel  be  a ditch,  or  canal,  of  20  feet  broad 
at  the  bottom,  22  at  the  top,  9 deep,  and  1000  feet  long  ; 

then  is  90  : 90  + x : : 20  : X 2 the  breadth  of  the 


surface  of  the  water  when  its  depth  in  the  canal  is  x ; and 
90  +x 

therefore  a = — - — - X 2000  is  the  surface  at  that  time. 


x9-2±fx 


is 


the 


• — Ax 

consequently  t or  — = 1100  .. 

^ 2 ay/gx  9 aVgx 

fluxion  of  the  time  ; the  correct  fluent  of  which,  when  x 

= 0,  is  1000  X 1000  X 186  X 3 = . 

9a  g 9 X Vg 

15459''§-  nearly,  or  4h.  17'  39"§,  beiDg  the  whole  time  of  ex- 
hausting by  a sluice  of  1 foot  square. 


PROBLEM  XVH. 


To  determine  the  Velocity  with  which  a Ball  is  discharged  from 
a Given  Piece  of  Ordnance,  with  a Given  Charge  of  Gun- 
powder. 

Let  the  annexed  figure 
. represent  the  bore  of  the 
gun  ;•  ad  being  the  part 
filled  with  gunpowder. 

And  put 

a = ab,  the  part  at  first  filled  with  powder  and  the  bag  ; 
h j==  ae,  the  whole  length  of  the  gunbore  ; 
c = -7834,  the  area  of  a circle  whose  diameter  is  1 ; 
d—  bd,  the  diameter  of  the  ball  : 

e — the  specific  gravity  of  the  ball,  or  weight  of  1 cubic  foot  ; 
\g=  16t2  feet,  descended  by  a body  in  1 second  ; 
m ~ 230  ounces,  the  pressure  of  the  atmosphere  on  a sq.  inch; 
n to  1 the  ratio  of  the  first  force  of  the  fired  powder,  to  the 
pressure  of  the  atmosphere  ; 
w=  the  weight  of  the  ball.  Also,  let 

r = Ac,  be  any  variable  distance  of  the  ball  from  a,  in 
moving  along  the  gunbarrel. 

Vor,.  IT.  31  First, 


D 


394 


PRACTICAL  EXERCISES  ON  FORCES. 


First,  cd2  is  = the  area  of  the  circle  bd  of  the  ball  ; 
there  vied2  is  the  pressure  of  the  atmosphere  on  bd  ; 
conseq  mned 2 is  the  first  force  of  the  powder  on  bd. 

But  the  force  of  the  inflamed  powder  is  proportional  to  its 
density,  and  the  density  is  inversely  as  the  space  it  fills  ; 
therefore  the  force  of  the  powder  on  the  ball  at  b,  is  to  the 
force  on  the  same  at  c,  as  ac  is  to  ab  ; that  is, 

x : a : : mned 2 : - — = f,  the  motive  force  at  c : 

x 

conseq.  - _ / the  accelerating  force  there. 

<w  IVX 

Hence,  theor.  10  of  forces  gives  v{,  = 2 gfx  = n‘nac-__  . 

the  fluent  of  which  is  v2  = ~g"  X hyp.  iog.  of  x. 

But  whenv  = 0,  then  x = a ; theref.  by  correction, 
v2  = X hyp.  log.  - is  the  correct  fluent ; conseq. 

v — < / Agmnacd^  ^ hyp.  log  -)  is  the  vel.  of  the  ball  at  c. 

and  v = */  mn  'cd-  X hyp.  log.  -)  the  velocity  with  which 

the  ball  issues  from  the  muzzle  at  f.  ; where  h denotes  the 
length  of  the  cylinder  filled  with  powder  ; and  a the  length  j 
to  the  hinder  part  of  the  ball,  which  will  be  more  than  h 
when  the  powder  does  not  touch  the  ball. 

Or,  by  substituting  the  numbers  for  g and  m,  and  chang- 
ing Ihe  hyperbolic  logarithms  for  the  common  ones,  then 

v X com.  log.|),  the  velocity  at  e,  infect. 

But.  the  content  of  the  ball  being  | cd3,  its  weight  is  - - 

■xsi  = L = • cc  — • which  being  substituted  for  xo, 

in  the  value  of  v,  it  becomes 

v — 2713  X com.  log.  — ),  the  velocity  at  e. 

When  the  ball  is  of  cast  iron;  takinge=7368,the  rule  becomes  s 

v = 100  ./  ( — 7 X log.  — ) for  the  veloc.  of  the  cast-iron  ball. 
\10a  a 

Or,  when  the  ball  is  of  lead  ; then 4 

v — 80f  </  X log.  — ) for  the  veloc.  of  the  leaden  ball- 

Corol  } 


PRACTICAL  EXERCISES  ON  FORCES. 


595 


Carol.  From  the  general  expression  for  the  velocity  v, 
above  given,  may  be  derived  what  mast  be  the  length  of  the 
charge  of  powder  u,  in  the  gun-barrel,  so  as  to  produce  the 
greatest  possible  velocity  in  the  ball  ; namely,  by  making  the 
value  of  v a maximum,  or,  by  squaring  and  omitting  the 


constant  quantities,  the  expression  a X hyp  log.  of  - 
a maximum,  or  its  fluxion  equal  to  nothing  ; that  is 


a X hyp.  log.  - — a = 0,  or  hyp.  log.  of  - = 1 ; hence 

= 2-71820,  the  number  whose  hyp.  log  is  1.  So  that 

a : b : : 1 : 2-71828,  or  as  4 to  11  nearly,  or  nearer  as  7 to 
19  ; that  is,  the  length  of  the  charge,  to  produce  the  great- 
est velocity,  is  the  x4Tth  part  of  the  length  of  the  bore,  or 
nearer  T7X  of  it 


But  by  actual  experiment- it  is  found,  that  the  charge  for  the 
greatest  velocity,  is  but  little  less  than  that  which  is  here 
computed  from  theory  ; as  may  be  seen  by  turning  to  page 
252  of  my  volume  of  Tracts,  where  the  corresponding  parts 
are  found  to  be,  for  four  different  lengths  of  gun,  thus,  T3¥, 
t32>  tbo  23o  * parts  here  varying,  as  the  gun  is  longer, 
which  allows  time  for  the  greater  quantity  of  powder  to  be 
fired,  before  the  ball  is  out  of  the  bore. 


SCHOLIUM. 


In  the  calculation  of  the  foregoing  problem,  the  value  of 
the  constant  quantity  n remains  to  be  determined.  It  denotes 
the  first  strength  or  force  of  the  fired  gunpowder,  just  before 
the  ball  is  moved  out  of  its  place.  This  value  is  assumed,  by 
Mr.  Robins,  equal  to  1000,  that  is,  1000  times  the  pressure 
of  the  atmosphere,  on  any  equal  spaces. 

But  the  value  of  the  quantity  n may  be  derived  much 
more  accurately,  from  the  experiments  related  in  my  Tracts, 
by  comparing  the  velocities  there  found  by  experiment,  with 
the  rule  for  the  value  of  v,  or  the  velocity,  as  above  com- 
puted by  theory,  viz.  - 

v = <00 y/{~  X log.  ofi),  or  = 100v/(^i  X log.  of*). 

Now,  supposing  that  v is  a given  quantity,  as  well  as  all  the 
other  quantities,  excepting  only  the  number,  n,  then  by  re- 
ducing this  equation,  the  Value  of  the  letter  n is  found  to  be 
as  follows,  viz.  


tlw  o 

n = ioo^  ^ com-  los-  of  *’  or 

when  h is  different  from  a. 


dw 

Iojo7i 


-f-  log. 


a 


Now 


396 


PRACTICAL  EXERCISES  ON  FORCES. 


Now,  to  apply  this  to  the  experiments.  By  page  240  of 
the  Tracts,  the  velocity  of  the  ball  of  1-96  inches  diameter, 
with  4 ounces  of  powder,  in  the  gun  No  1,  was  1100  feet 
per  second  : and,  by  pa.  494,  vol  1 , the  length  of  the  gun,  when 
corrected  for  the  spheroidal  hollow  in  the  bottom  of  the  bore, 
was  28-53  ; also,  by  page  228.  the  length  of  the  charge, 
when  corrected  in  like  manner,  was  3 45  inches  of  powder 
and  bag  together,  but  2-54  of  powder  only  : so  that  the 
values  of  the  quantities  in  the  rule,  are  thus  : a = 3-45  ; b 
= 28-53  ; d = 1-96  ; h — 2-54  ; and  v — 1100  : then,  by 
substituting  these  values  instead  of  the  letters,  in  the  theorem 

n = com.  log.  of  it  comes  out  n — 750,  when 

h is  considered  as  the  same  as  a.  And  so  on,  for  the  other 
experiments  there  treated  of. 

It  is  here  to  be  noted  however,  that  there  is  a circum- 
stance in  the  experiments  delivered  in  the  Tracts,  just  men- 
tioned, which  will  alter  the  value  of  the  letter  a in  this 
theorem,  which  is  this,  viz.  that  a denotes  the  distance  of 
the  shot  from  the  bottom  of  the  bore  ; and  the  length  of  the 
charge  of  powder  alone  ought  to  be  the  same  thing  ; but,  in 
the  experiments,  that  length  included,  besides  the  length  of 
real  powder,  the  substance  of  the  thin  flannel  bag  in  which 
it  was  always  contained,  of  which  the  neck  at  least  extended 
a considerable  length,  being  the  part  where  the  open  end  was 
wrapped  and  tied  close  round  with  a thread.  This  circum- 
stance causes  the  value  of  n,  as  found  by  the  theorem  above, 
to  come  out  less  than  it  ought  to  be  for  it  shows  the  strength 
of  the  inflamed  powder  when  just  fired,  and  when  the  flame 
fills  the  whole  space  a before  occupied  both  by  the  real  pow- 
der and  the  bag,  whereas  it  ought  to  show  the  first  strength 
of  the  flame  when  it  is  supposed  to  be  contained  in  the  space 
only  occupied  by  the  powder  alone,  without  the  bag.  The 
formula  will  therefore  bring  out  the  value  of  n too  little,  in 
proportion  as  the  real  space  filled  by  the  powder  is  less  than 
the  space  filled  both  by  the  powder  and  its  bag.  In  the  same 
proportion  therefore  must  we  increase  the  formula,  that  is, 
in  the  proportion  of  h,  the  length  of  real  powder,  to  a the 
length  of  powder  and  bag  together.  When  the  theorem  is 

so  corrected,  it  becomes  -f-  com.  log.  of  — . 

Now,  by  pa.  228  of  the  Tracts,  there  are  given  both  the 
lengths  of  all  the  charges,  or  values  of  a.  including  the  bag, 
and  also  the  length  of  the  neck  and  bottom  of  the  bag.  which 
is  0 91  of  an  inch,  which  therefore  must  be  subtracted  from 
; • ' all 


PRACTICAL  EXERCISES  ON  FORCES.  397 


all  the  values  of  a,  to  give  the  corresponding  values  of  h. 
This  in  the  example  above  reduces  3-45  to  2-54. 

Hence,  by  increasing  the  above  result  750,  in  proportion 
i of  2-54  to  3-45,  it  becomes  1018.  And  so  on  for  the  other 
experiments. 

But  it  will  be  best  to  arrange  the  results  in  a table,  with 
the  several  dimensions,  when  corrected,  from  which  they  are 
computed,  as  here  below. 


Table  of  Velocities  of  Balls  and  First  Force  of  Powder,  4’C 


Gun. 

Charge  of  Powder.  | 

Velocity 
or  value 

of  V. 

First 
force,  or 
value  of 
n. 

No. 

Length, 
or  value 
of  b. 

Weight 

in 

ounces. 

Leng 
va 
ef  a. 

th  or 
ue 
of  h. 

inches 

4 

3-45 

2-54 

1100 

1018 

1 

28-53 

8 

5-99 

5-08 

1430 

1164 

16 

11-07 

10-16 

1430 

967 

4 

3-45 

2-54 

1180 

1077 

2 

38-43 

8 

5-99 

5-08 

1580 

1193 

16 

11-07 

10-16 

1660 

984 

4 

3-45 

2-54 

1300 

1067 

o 

O 

57-70 

8 

5-99 

5:08 

1790 

1256 

- . 

16 

11-07 

10-16 

2000 

1076 

4 

3-45 

2-54 

1370 

1060 

4 

80-23 

8 

5-99 

5-08 

1940 

1289 

16 

11  07 

1016 

2200 

1085 

Where  it  may  be  observed,  that  the  numbers  in  the  column 
)f  velocities,  U430  and  2200,  are  a little  increased,  as,  from  a 
view  of  the  table  of  experiments,  they  evidently  required  to 
be.  Also  the  value  of  the  letter  d is  constantly  1-96  inch. 

Hence  it  appears,  that  the  value  of  the  letter  n,  used  in 
he  theorem,  though  not  yet  greatly  different  from  the  num- 
ber 1000,  assumed  by  Mr.  Robins,  is  rather  various,  both  for 
;he  different  lengths  of  the  gun,  and  for  the  different  charges 
vith  the  same  gun. 


But 


PRACTICAL  EXERCISES  ON  FORCES. 


398 


But  this  diversity  in  the  value  of  the  quantity  n,  or  the  first 
force  of  the  inflamed  gunpowder,  is  probably  owing  in  some 
measure  to  the  omission  of  a material  datum  in  the  calculation 
of  the  problem,  namely,  the  weight  of  the  charge  of  powder, 
which  has  not  all  been  brought  into  the  computation.  For  it 
is  manifest,  that  the  elastic  fluid  has  not  only  the  ball  to  move 
and  impel  before  it,  but  its  own  weight  of  matter  also.  The 
computation  may  therefore  be  renewed,  in  the  ensuing  pro- 
blem, to  take  that  datum  into  the  account. 


PROBLEM  XVIII. 


To  determine  the  same  as  in  the  lost  Problem;  taking  both  the 
Weight  of  Powder  and.  the  Ball  into  the  Calculation. 


Besides  the  notation  used  in  the  last  problem,  let  Vp  denote 
the  weight  of  the  powder  in  the  charge,  with  the  flannel  bag 
in  which  it  was  inclosed. 

Now,  because  the  inflamed  powder  occupies  at  all  times  the 
part  of  the  gun  bore  which  is  behind  the  ball,  its  centre  of 
gravity,  or  the  middle  part  of  the  same,  will  move  with  only 
half  the  velocity  that  the  ball  moves  with  ; and  this  will  require 
the  same  force  as  half  the  weight  of  the  powder,  kc.  moved 
with  the  whole  velocity  of  the  ball.  Therefore,  in  the  con- 
clusion derived  in  the  last  problem,  we  are  now,  instead  of  w, 
to  substitute  the  quantity  p -f-  w ; and  when  that  is  done  the 

last  velocity  will  come  out,  v = (' — — X com.  log.—.) 


P+w 

And  from  this  equation  is  found  the  value  of  n,  which 


— log.  of 


> P + m , , r b , , 

. -,  ==- r“  -f-  log.  of  — , bv  sub- 

2'230/ui2  0 a 8507  h 3 a 

s titrating  for  d its  value  T96,  the  diameter  of  the  ball. 

Now  as  to  the  ball,  its  medium  weight  was  16  oz.  13  dr  = 
1 6 ■ C 1 oz.  Ami  the  weights  of  the  bags  containing  the  several 

charges  of  powder,  viz  4 oz  8 oz.  16  oz  were  8 dr.  12  dr. 
and  1 oz.  5 dr  ; then  adding  these  to  the  respective  contained 
weights  of  powder,  the  sums,  4-5  oz.  8-75  oz.  17  31  oz.  are 
the  values  of  tip,  or  the  weights  of  the  powder  and  bags  ; 
the  halves  of  which,  or  2-25,  and  4-38,  and  8 66,  are  the 
values  of  the  quantity  p for  those  three  charges  ; and  these 
being  added  to  1 6 • C I , the  constant  weight  of  the  ball,  there 
are  obtained  the  three  values  of  p -f-  rv  for  the  three  charges 
of  powder,  which  values  therefore  are  19-06  oz.  and  21-19 
oz.  and  25-47  oz.  Then,  by  calculating  the  values  of  the 
irst  force  n,  by  the  last  rule  above,  with  these  new  data,  the 
whole  will  be  found  as  in  the  following  table. 

The 


PRACTICAL  EXERCISES  ON  FORCES. 


39? 


The  Gun. 

Charge  of  Powder,  i 

Weight  of 
ball  and 
charge,  or 
values  of 
p-j-sy 

Velocity, 
or  the 
values 
of  V. 

First 
force 
or  the 
value 
of  n. 

; No 

Length 
or  value 
of  b. 

Weight 

in 

ounces. 

Leng 
val 
of  a. 

h or  ; 
ue  i 
of  h.- 

inches 

4 

3 45 

2-54 

19  06 

1100 

1 155 

1 

28-53 

8 

5-99 

5 08 

21-19 

1430 

1470 

16 

11-07 

10-16 

25-47 

1430 

1456| 

4 

3"45 

2-54 

19-06 

1 180 

1167 

2 

38-43 

8 

5-99 

5 08 

21-19 

1 580 

150b 

16 

1 1 07 

10  16 

25-47 

1660 

1492 

* 

4 

3 45 

2-54 

19  06 

1300 

1210 

! 3 

57-70 

8 

5-99 

5-08 

21  19 

3790 

1586 

16 

1 1 07 

10-16 

25-47 

2000 

1646 

- 

4 

3-45 

2-54 

19-06 

1370 

1203 

4 

80-23 

8 

5 99 

5-08 

£1-19 

1940 

1627 

16 

! 1 07 

10-16 

25-47 

2200 

1648 

And  here  it  appears  that  the  values  of  n,  the  first  force  of 
the  charge  are  much  more  uniform  and  regular  than  by  the 
former  calculations  in  the  preceding  problem  at  least  in  all 
excepting  the  smallest  charge,  4 oz  in  each  gun  ; which  it 
would  seem  must  be  owing  to  some  general  cause  or  causes. 
Nor  have  we  long  to  search,  to  find  out  what  those  causes 
may  be.  For  when  it  is  considered  that  these  numbers  for 
the  value  of  n,  in  the  last  column  of  the  table,  ought  to  ex- 
hibit the  first  force  of  the  fired  powder,  when  it  is  supposed 
to  occupy  the  space  only  in  which  the  bare  powder  itself 
; lies  and  that  whereas  it  is  manifest  that  the  condensed  fluid 
of  the  charge  in  these  experiments,  occupies  the  whole 
space  between  tiie  ball  and  the  bottom  of  the  gun  bore,  or 
the  whole  space  taken  up  by  the  powder  and  the  bag  or  car- 
tridge together,  which  exceeds  the  former  space,  or  that  of 
the  powder  alone,  at  least  in  the  proportion  of  the  circle  of 
the  gun  bore,  to  the  same  as  diminished  by  the  thickness  ot 
the  surrounding  flannel  of  the  bag  that  contained  the  pow- 
der ; it  is  manifest  that  the  force  was  diminished  on  that  ac- 
count. Now  by  gently  compressing  a number  of  folds  of 
the  flannel  together,  it  lias  been  found  that  the  thickness  of 
the  single  flannel  was  equal  to  the  40th  part  of  an  inch  ; the 
I double  of  which,  or  -05  of  an  inch,  is  therefore  the 

quantity 


400 


PRACTICAL  EXERCISES  ON  FORCES. 


quantity  by  which  the  diameter  of  the  circle  of  the  powder 
within  the  bag,  was  less  than  that  of  the  gun  bore  But  the  i 
diameter  of  the  gun  bores  was  2-02  inches ; therefore,  deduct- 
ing the  '05,  the  remainder  1-97  is  the  diameter  of  the  powder 
cylinder  within  the  bag  : and  because  the  areas  of  circles  are 
to  each  other  as  the  spaces  of  their  diameters,  and  the  squares 
of  these  numbers,  T97  and  2-02,  being  to  each  other  as  388  to 
408,  or  as  97  to  102;  therefore,  on  this  account  ( alone,  the 
numbers  before  found,  for  the  value  of  n,  must  be  increased 
in  the  ratio  of  97  to  102. 

But  there  is  yet  another  circumstance,  which  occasions 
the  space  at  first  occupied  by  the  inflamed  powder  to  be 
larger  than  that  at  which  it  has  been  taken  in  the  foregoing  cal-  1 
culations,  and  that  is  the  difference  between  the  content  of  a i 
sphere  and  cylinder.  For  the  space  supposed  to  be  occupied 
at  first  by  the  elastic  fluid,  was  considered  as  the  length  of  a i 
cylinder  measured  to  the  hinder  part  of  the  curve  surface 
of  the  ball,  which  is  manifestly  too  little  by  the  difference 
■s  between  the  content  of  half  the  ball  and  a cylinder  of  the  same 
length  and  diameter,  that  is,  by  a cylinder  whose  length  is  a j 
the  semidiameter  of  the  ball.  Now  that  diameter  was  1-96  , 
inches  ; the  half  of  which  is  0-98,  and  a of  this  is  0-33  near- 
ly. Hence  then  it  appears  that  the  lengths,  of  the  cylinders 
at  first  filled  by  the  dense  fluid,  viz.  3-45s  and  5-99,  and. 
11-07,  have  been  all  taken  too  little  by  0 33  ; and  hence  it  fol-  I 
lows  that,  on  this  account  also,  all  the  numbers  before  found  j 
for  the  value  of  the  first  force  n,  must  be  further  increased  in 
the  ratios  of  3-45  and  5 99  and  11  07,  to  the  same  numbers 
increased  by  0-33,  that  is,  to  the  numbers  3-78  and  6-32  anil 
11  40. 

Compounding  now  these  last  ratios  with  the  foregoing 
one,  viz.  97  to  102,  it  produces  these  three,  viz.  the  ratios 
of  334  and  581  and  1074,  respectively  to  385  and  647  and 
1163.  Therefore,  increasing  the  last  column  of  numbers, 
for  the  value  of  n,  viz.  those  of  the  4 oz.  charge  in  the  ratio 
of  334  to  385,  and  those  of  the  8 oz.  charge  in  the  ratio  of 
581  to  647,  and  those  of  the  i 
16  oz.  charge  in  the  ratio  of 
1074  to  1163,  with  every  gun, 
they  will  be  reduced  to  the 
numbers  in  the  annexed  ta- 
ble ; where  the  numbers  are 
still  larger  and  more  regular 
than  before. 


Powder. 

The  Guns. 

oz. 

1 

2 

3 

4 

4 

1372 

1387 

1438 

1430 

8 

1637 

1677 

1766 

1812 

16 

1577 

1616 

1782 

1784 

Thus 


PRACTICAL  EXERCISES  ON  FORCES. 


40^ 


Thus  then  at  length  it  appears  that  the  first  force  of  the 
inflamed  gunpowder,  when  occupying  only  the  space  at  first 
filled  with  the  powder,  is  about  1800.  that  is  1800  times 
fhe  elasticity  of  the  natural  air,  or  pressure  of  the  atmosphere 
in  the  charges  with  8 oz.  and  16  oz.  of  powder,  in  the  two 
longer  guns  ; but  somewhat  less  in  the  two  shorter,  probably- 
owing  to  the  gradual  firing  of  gunpowder  in  some  degree  ; 
and  also  less  in  the  lowest  charge  4 oz.  in  all  the  guns,  which 
may  probably  be  owing  to  the  less  degree  of  heat  in  the  small 
charge.  But  besides  the  foregoing  circumstances  that  have 
been  noticed,  or  used  in  the  calculations,  there  are  yet  several 
others  that  might  and  ought  to  be  taken  into  the  account,  in  or- 
der to  a strict  and  perfect  solution  of  the  problem ; such  as, 
the  counter  pressure  of  the  atmosphere,  and  the  resistance  of 
the  air  on  the  fore  part  of  the  ball  while  moving  along  the 
bore  of  the  gun  ; the  loss  of  the  elastic  fluid  by  the  vent  and 
windage  of  the  gun  ; the  gradual  firing  of  the  powder  ; the 
unequal  density  of  the  elastic  fluid  in  the  different  parts  of  the 
space  it  occupies  between  the  ball  and  the  bottom  of  the  bore  ; 
fhe  difference  between  pressure  aud  percussion  when  the  ball 
is  not  laid  close  to  the  powder  ; and  perhaps  some  others  : on 
all  which  accounts  it  is  probable  that  instead  of  1800,  the 
first  force  of  the  elastic  fluid  is  not  less  than  2000  times  the 
strength  of  natural  air. 


Corol.  From  the  theorem  last  used  for  the  velocity  of  th§ 

ball  and  elastic  fluid  viz.  v = */  ( ~~o0h  ‘ .n  q.wi-  == 

v + w 65  a 


-rr  log  -),  we  may  find  the  velocity  of  the  elas- 
tic fluid  alone,  viz.  by  taking  w,  or  the  weight  of  the  ball, 
= 0 in  th  e theorem,  by  which  it  becomes  barely  v = 


y/ -i-  log.-),  for  that  velocity. 


And  by  computing 


the  several  preceding  examples  by  this  theorem,  supposing 
the  value  of  n to  be  2000,  the  conclusions  come  out  a little 
various,  being  between  4000  and  5000,  but  most  of  them  nearer 
to  the  latter  number.  So  that  it  may  be  concluded  that  the 
velocity  of  the  flame,  or  of  the  fired  gunpowder  expands  it- 
self at  the  muzzle  of  the  gun,  at  the  rate  of  about  5000  feet 
per  second  nearly, 


ON 


6% 


402 


MOTION  OF  BODIES  IN  FLUIDS, 


ON  THE  MOTION  OF  BODIES  IN  FLUIDS, 
PROBLEM  XIX. 


To  determine  the  Force  of  Fluids  in  Motion;  and  the  Circum- 
stances attending  Bodies  Moving  in  Fluids. 


1.  It  is  evident  that  the  resistance  to  a plane,  moving 
perpendicularly  through  an  infinite  fluid,  at  rest,  is  equal  to 
the  pressure  or  force  of  the  fluid  on  the  plane  at  rest,  and 
the  fluid  moving  with  the  same  velocity,  and  in  the  contrary 
direction,  to  that  of  the  plane  in  the  former  case.  But  the 
force  of  the  fluid  in  motion,  must  be  equal  to  the  weight  or 
pressure  which  generates  that  motion  ; and  which,  it  is 
known,  is  equal  to  the  weight  or  pressure  of  a column  of 
the  fluid,  whose  base  is  equal  to  the  plane,  and  its  altitude 
equal  to  the  height  through  which  a body  must  fall  by  the 
force  of  gravity,  to  acquire  the  velocity  of  the  fluid  : and 
that  altitude  is,  for  the  sake  of  hrevit}',  called  the  altitude 
due  to  the  velocity.  So  that,  if  a denote  the  area  of  the 
plane,  v the  velocity,  and  n the  specific  gravity  of  the  fluid  ; 


V2 

then  the  altitude  due  to  the  velocity  v being  ^-,the  whole 


resistance,  or  motive  force  m,  will  be  a X n X — =— - ; g 

4?  4g  5 

being  16  tl  feet.  And  hence,  cscteris  parilus,  the  resistance 
is  as  the  square  of  the  velocity. 


2.  This  ratio  of  the  square  of  the  velocity,  may  be  other- 
wise derived  thus.  The  force  of  the  fluid  in  motion,  must 
be  as  the  force  of  one  particle  multiplied  by  the  number  of 
them  ; but  the  force  of  a particle  is  as  its  velocity  ; and  the 
number  of  them  striking  the  plane  in  a given  time,  is  also  as 
the  velocity  ; therefore  the  whole  force  is  as  v X v or  r2, 
that  is  the  square  of  the  velocity. 

3.  If  the  direction  of  motion,  instead  of  being  perpendi- 
cular to  the  plane,  as  above  supposed,  be  inclined  to  it  in 
any  angle,  the  sine  of  that  angle  being  s to  the  radius  1 : 
then  the  resistance  to  the  plane,  or  the  force  of  the  fluid 

against 


MOTION  OF  BODIES  IN  FLUIDS. 


403 


s against  the  plane,  in  the  direction  of  the  motion,  as  assigned 
! above,  will  be  diminished  in  the  triplicate  ratio  of  radius  to 
the  sine  of  the  angle  of  inclination,  or  in  the  ratio  of  1 to  s3. 
t For  AB  being  the  direction  of  the  plane, 
and  bd  that  of  the  potion  making  the 
angle  abd,  whose  sine  is  s ; the  number 
of  particles,  or  quantity  of  the  fluid 
striking  the  plane,  will  be  diminished  in 
the  ratio  of  I to  s,  or  of  radius  to  the 
sine  of  the  angle  b of  inclination  ; and 
the  force  of  each  particle  will  also  be  diminished  in  the  same 
ratio  of  1 to  s : so  that  on  both  these  accounts,  the  whole  re- 
sistance will  be  diminished  in  the  ratio  of  1 to  s2,  or  in  the 
duplicate  ratio  of  radius  to  the  sine  of  the  said  angle.  But 
again,  it  is  to  be  considered  that  this  whole  esistance  is  ex- 
erted in  the  direction  be  perpendicular  to  the  plane  ; and  any 
force  in  the  direction  be,  is  to  its  effect  in  the  direction  ae, 
parallel  to  bd,  as  ae  to  be,  that  is  a*  1 to  s.  So  that  finally, 
on  all  these  accounts,  the  resistance  in  the  direction  of  motion, 
is  diminished  in  the  ratio  of  1 to  s3,  or  in  the  triplicate  ratio  of 
radius  to  the  sine  of  inclination.  Hence,  comparing  this 
with  article  1,  the  whole  resistance,  or  the  motive  force  on  the 

plane,  will  be  m = " . 

4.  Also,  if  w denote  the  weight  of  the  body,  whose 
plane  face  a is  resisted  by  the  absolute  force  rp  ; then  the 

retarding  force  f,  or  ™ will  be  ~ 

° J v 4 gw 


5.  And  if  the  body  be  a cylinder,  whose  face  or  end  is  a, 
and  diameter  d,  or  radius  r,  moving  in  the  direction  of  its 
axis  ; because  then  s = 1,  and  a = pr2  = \pd2 , where  p — 
3-1416:  the  resisting  force  m will  be  - 


npd2v2 npr 2 v 2 

I6g  ~ 4 g 


, and  the  retarding  force / = 


npd2  ‘ rpr2 
1 6gw  4 g-w 


6.  This  is  the  value  of  the  resistance  when  the  end  of  the 
cylinder  is  a plane  perpendicular  to  its  axis,  or  to  the  direc- 
tion of  motion.  But  were  its  face  a conical  surface,  or  an 
elliptic  section,  or  any  other  figure  every  where  equally  in- 
clined to  the  axis,  the  sine  of  inciination  being  s : then  the 
number  of  particles  of  the  fluid  striking  the  face  being  still 
the  same  but  the  force  of  eaeh,  opposed  to  the  direction 


of 


404 


MOTION  OF  BODIES  IN  FLUIDS. 


of  motion,  diminished  in  the  duplicate  ratio  of  radius  to  the  - 
sine  of  inclination,  the  resisting  force  rn  would  be 

7i/)dzv^n2  vpr-v-  s2 

16§-  4g 

But  if  the  body  were  terminated  by  an  end  or  face  of  any  I 
other  form  as  a spherical  one,  or  such  like,  where  every  part  ! 
of  it  has  a different  inclination  to  the  axis  ; then  a further  j 
investigation  becomes  necessary,  such  as  in  the  following  pro- 
position. 

PROBLEM  XX. 

To  determine  the  Resistance  of  a Fluid  to  any  Body,  moving  in 
it,  of  a Curved  End  ; as  a Sphere,  or  a Cylinder  with  a Hem~ 
ispherical  End,  &rc. 

1.  Let  bead  be  a section  through  the 
axis  ca  of  the  solid,  moving  in  the  direc- 
tion of  that  axis.  To  any  point  of  the 
curve  draw  the  tangent  eg,  meeting  the 
axis  produced  in  g : also,  draw  the  per- 
pendicular ordinates  ef,  ef,  indefinitely 
near  each  other  ; and  draw  ae  parallel  to 
CG. 


Putting  cf  — x , ef  ==  y,  be  = z,  s = sine  Z.  u to  ra- 
dius l,andp,  = 3-1416  : then  %py  is  the  circumference  whose 
radius  is  ef,  or  the  circumference  described  by  the  point  e, 
in  revolving  about  the  axis  ca  ; and  2 py  X Ee  or  2 pyz  is  the 
fluxion  of  the  surface,  or  it  is  the  surface  described  by  Ee, 
in  the  said  revolution  about  ca  , and  which  is  the  quantity 
represented  by  a in  art.  3 of  the  last  problem  : hence 

72*j2  $ 3 „ pnu 2 $3  ...  . . 

— — X Hpyz  or  — X yz  is  the  resistance  on  that  ring, 

or  the  fluxion  of  the  resistance  to  the  body,  whatever  the 
figure  of  it  may  be.  And  the  fluent  of  which  will  be  the  re- 
sistance required. 


2.  In  the  case  of  a spherical  form  : putting  the  radius  ca 

, . — o ; ef  cf  x 

or  cb  = r,  we  have  y = +/  r2  — x- , s = — — : and 
* v eg  ce  r 

yz  , or  ef  X ec  — ce  X ae  — rx  ; therefore  the  general 
fluxion -'v  X s3yz  becomes^-  X X r x X x3x  } 


% 


‘8r~ 


the 


MOTION  OF  BODIES  IN  FLUIDS. 


405 


ihe  fluent  of  which,  or  ^ — - x 4,  is  the  resistance  to  the 

8 gr* 

spherical  surface  generated  by  be.  And  when  x or  cf  is  = r 
or  ca,  it  becomes  ■ --  for  the  resistance  on  the  whole 

85- 

hemisphere  ; which  is  also  equal  to  » where  d — 2r 

the  diameter. 


3.  But  the  perpendicular  resistance  to  the  circle  of  the 
same  diameter  d or  bd,  by  art.  5 of  the  preceding  problem, 

f)7l"V ^ (l^  , 

is  — - — ; which,  being  double  the  former,  shows  that  the 

resistance  to  the  sphere,  is  just  equal  to  half  the  direct  resist- 
ance to  a great  circle  of  it,  or  to  a cylinder  of  the  same  di- 
ameter. 


4.  Since  }pd3  is  the  magnitude  of  the  globe  ; if  n denote 
its  density  or  specific  gravity,  its  weight  w will  be  = }pdsN, 

and  therefore  the  retardive  force  f or  - = p ,“'2  - X — — 

J W 32g  />N£?S 

= — • ; which  is  also  = — — by  art.  8 of  the  general 

16gvd  4 gs  J a 

theorems  in  page  380  ; hence  then  and  s = — 

X | d ; which  is  the  space  that  would  be  described  by  the 
globe,  while  its  whole  motion  is  generated  or  destroyed  by  a 
constant  force  which  is  equal  to  the  force  of  resistance,  if  no 
other  force  acted  on  the  globe  to  continue  its  motion.  And  if 
the  density  of  the  fluid  were  equal  to  that  of  the  globe,  the 
resisting  force  is  such,  as,  acting  constantly  on  the  globe  with- 
out any  other  force,  would  generate  or  destroy  its  motion  in 
describing  the  space  f d,  or  a of  its  diameter,  by  that  accelerat- 
ing or  retarding  force. 

5 Hence  the  greater  velocity  that  a globe  will  acquire 
by  descending  in  a fluid,  by  means  of  its  relative  weight  in 
the  fluid,  will  be  found  by  making  the  resisting  force  equal 
to  that  weight.  For,  after  the  velocity  is  arrived  at  such  a 
degree,  that  the  resisting  force  is  equal  to  the  weight  that 
urges  it,  it  will  increase  no  longer,  and  the  globe  will  after- 
wards continue  to  descend  with  that  velocity  uniformly. 
Now,  n aod  n being  the  separate  specific  gravities  of  the 
globe  and  fluid,  n n will,  be  the  relative  gravity  of  the 
globe  in  the  fluid,  and  therefore  w — }pd3  (n— n)  is  the 

weight 


406 


MOTION  OF  BODIES  IN  FLUIDS. 


weight  by  which  it  is  urged  ; also  m - - 

— is  the  resistance;  consequently^^--  = ±pd3(rs—n.j 
when  the  velocity  becomes  uniform  : from  which  equation 
is  found  v = y/  (4^  . id  . - — -),  for  the  said  uniform  or 
greatest  velocity. 

And,  by  comparing  this  form  with  that  in  art.  6 of  the  gene- 
ral  theorems  in  page  379,  it  will  appear  that  its  greatest  velo-  , 
locity,  is  equal  to  the  velocity  generated  by  the  accelerat- 
ing force  in  describing  the  space  %d,  or  equal  to  the 

velocity  generated  by  gravity  in  freely  describing  the  space 
1 X -*-d.  If  n — 2 n,  or  the  specific  gravity  of  the  !' 

globe  be  double  that  of  the  fluid,  then  JLTF  = l = the 

* n 

natural  force  of  gravity  ; and  then  the  globe  will  attain  its 
greatest  velocity  in  describing  ±d  or  f of  its  diameter.  — It 
is  further  evident,  that  if  the  body  be  very  small,  it  will 
very  soon  acquire  its  greatest  velocity,  whatever  its  density 
may  be. 

Exam.  If  a leaden  ball,  of  1 inch  diameter,  descend  in 
water,  and  in  air  of  the  same  density  as  at  the  earth’s  surface, 
the  three  specific  gravities  being  as  1 li,  and  1,  and  I 

Than  v = 4 . 16T7  . . JU^  = 193  = 8-5944  feet, 

is  the  greatest  velocity  per  second  the  ball  can  acquire  by 
descending  in  water.  And  v — y/  4 . >Ty  . I 

nearly  = 5g°  y/  34"-’-u  3 = 259-52  is  the  greatest  velocity  it  can 
acquire  in  air. 

But  if  the  globe  were  only  Ti-7  of  an  inch  diameter,  the 
greatest  velocities  it  could  acquire,  would  be  only  -J-  of  these, 
namely  T*767  of  a foot  in  water,  and  26  feet  nearly  in  air.  And 
if  the  ball  were  still  further  diminished,  the  greatest  velocity 
would  also  be  diminished,  and  that  in  the  subduplicate  ratio 
of  the  diameter  of  the  ball. 

TIIOBLEM  XXI. 

To  determine  the  Relations  of  Velocity,  Space,  and  Time , of  a 
Ball  moving  in  a Fluid,  in  which  it  is  projected  with  a Given 


MOTION  OF  BODIES  IN  FLUIDS. 


407 


1.  Let  a = the  first  velocity  of  projection,  x the  space 
described  in  any  time  t , and  v the  velocity  then.  Now,  by- 

art.  4 of  the  last  problem,  the  accelerative  force  f — 

r J \6gxd 

where  n is  the  density  of  the  fluid,  n that  of  the  ball,  and 

d its  diameter.  Therefore  the  general  equation  vi  = ~gf  's 

becomes  vy  — - 

~TV~,  x ? and  hence  - = ~~--x  = — bx,  putting  b for 
8 Nd  v Ssd  x ° 8 sat 

The  correct  fluent  of  this,  is  log.  a — log.  v or  log.  - = bx. 

Or,  putting  c = 2-71 828 1 828,  the  number  whose  hyp.  log. 

is,  1.  then  is  - = c *,  and  the  velocity  v ~ ~ = cc-bx. 

2.  The  velocity  v-  at  any  time  being  the  c~bi  part  of  the 
first  velocity,  therefore  the  velocity  lost  in  any  time,  will  be 

- cbx — J 

the  1 — c~bx  part,  or  the part  of  the  first  velocity. 


EXAMPLES. 


Exam.  1.  If  a globe  be  projected,  with  any  velocity,  in  a. 
medium  of  the  same  density  with  itself,  and  it  describe  a 
space  equal  to  3 d or  3 of  its  diameters.  Then  x = 3d,  and 


3n  ^ rbx—  I 

b — — =—  therefore  bx  = f,  and  — - — = 

- 8 ltd  8d  8 cbx 

velocity  lost,  or  nearly  § of  the  projectile  velocity. 


2-08 

3<f8 


is 


the 


Exam.  2.  If  an  iron  ball  of  2 inches  diameter  were  pro- 
jected with  a velocity  of  1200  feet  per  second  : to  find  the 
velocity  lost  after  moving  through  any  space,  as  suppose 
500  feet  of  air  : we  should  have  d = T2?  ~ «>  a = 1200,  x 
— 500,  n = 7^-,  n = -0012  ; and  therefore  bx  = 


8nx  _ 3 . 12 . 500 . 3 . 6 _ 81  _ 1200 

%sd  8 . 22  . 10000  440  ’ 30  * ~ JU_ 


998  feet  per 


second  : having  lost  202  feet,  or  nearly  i of  its  first  velo- 
city. 


Exam.  3.  If  the  earth  revolved  about  the  sun,  in  a me- 
dium as  dense  as  the  atmosphere  near  the  earth’s  surface  ; 
?nd  were  required  to  find  the  quantity  of  motion  lost  in  a 


year- 


408 


MOTION  OF  BODIES  IN  FLUIDS; 


year.  Then  since  the  earth’s  mean  density  is  about  4|,  and 
its  distance  from  the  sun  12000  of  its  diameters,  we  have 
24000  X 3 1416  = 75398  diameters  = x,  and  bx  — - - - II 

370398  • = 7-5398  ; hence  = jfff  parts  are 


8,10000.9  ’ eta 

lost  of  the  first  motion  in  the  space  of  a year,  and  only  the 
TsVt  Part  remains. 


Exam.  4.  If  it  be  required  to  determine  the  distance 
moved,  x,  when  the  globe  has  lost  any  part  of  its  motion,  as  , 
suppose  i and  the  density  of  the  globe  and  fluid  equal  ; I 

The  general  equation  gives  x — ^ X log.  — = -j-  X log  of  . 

2 = l-8483925d  So  that  the  globe  loses  half  its  motion  be-  1 
fore  it  has  described  twice  its  diameter. 


3.  To  find  the  time  t ; we  have  1 — ~ — 


Now  to  find  the  fluent  of  this,  put  z = c<k  ; then  is  bx  = 

i .7  z . i • cbxi 

log.  z,  and  bjr  = — , or  x 1 conseq  * or  — — = 

z bz  a 


Zx  z , , . z 

— = — - and  hence  t — — = — . 

a ab  ab  ab 


But  as  t and  x vanish 


together,  and  when  x 
therefore,  by  correction,  t 
the  time  sought  ; where  b 


0, 


the  quantity  c~  is  = -Ji 


1 

bv 


1 

ba 


c*  - ii 


8n  d 


— , and  v — — the  velocity, 

jrr  cbx 


Exam.  If  an  iron  ball  of  2 inches  diameter  were  projected 
in  the  air  with  a velocity  of  1200  feet  per  second  ; and  it 
were  required  to  determine  in  what  time  it  would  pass  over 
500  yards  or  1500  feet,  and  what  would  be -its  velocity  at  the 
end  of  that  time  : We  should  have,  as  in  exam.  2 above, 


b = 

1 

b 


12 


1 

2716’ 

1 

1 7 a 1200’ 

Consequently  v 


and  bx  =■ 


1500 


375 

8 . 12  . 10000  . r 2716’  “““  — 2716  M 679  ’ 

2716  and  1 ~ — and  - = — = 1-7372  1 

? ~ i onn’  n 


hence 


near-' 


v a 1200  690 

ly.  Consequently  v = 690  is  the  velocity  ; and  t = 

■— (- — ) =2716X(-^- — ) = 1|4  seconds,  is  the  time 

bKv  a ' '690  1200'  *b 

required,  or  1"  and  f nearly. 


PROBLEM 


MOTION  OF  BODIES  IN  FLUIDS. 


409 


PROBLEM  XXII. 


To  determine  the  Relations  of  Space,  Time,  and  Velocity,  when 
a Globe  descends,  by  its  own  Weight,  in  a Fluid. 

The  foregoing  notation  remaining,  viz.  d = diameter, 
N and  n the  density  of  the  ball  and  fluid,  and  v,  s,  t,  the 
velocity,  space,  and  time,  in  motion  ; we  have  }pd3  = the 
magnitude  of  the  ball,  and  %pd3  (n — n)  — its  weight  in  the 

_ pnd  v _ jjg  resjstailce  from  the  fluid  ; 
pnd2v2 


fluid,  also  m 

32g 

eonsequently  }pd3  (n — n ) 


32g- 


is  the  motive  force  by 


which  the  ball  is  urged  ; which  being  divided  by  } ud3,  the 
quantity  of  matter  moved,  gives  f = 1 — for  the 


accelerative  force. 


2.  Hence  v'v  — 2gfs,  and 's  — ——  — 

2§j 


c , . 3n 

2g  (n  — n)—  —v 


8 d 


, putting  b = £ andi  = 


3n 


— ys  ^ ^ _ 

b a — u2’  *’fe  " 8n d'  ~ ~ a 2g  • 8d  (n  — n ) 

or  ab  = 2 g nearly  ; the  fluent  of  which  is  s = - - - - 

^r-  X log.  of  — — an  expression  for  the  space  s,  in  terms 

dO  CL 

of  the  velocity  v.  That  is,  when  s and  v begin,  or  are  equal 
to  nothing,  both  together. 


But  if  the  body  commence  motion  in  the  fluid  with  a cer- 
tain given  velocity  e,  or  enter  the  fluid  with  that  velocity, 
like  as  when  the  body,  after  falling  in  empty  space  from  a 
certain  height,  falls  into  a fluid  like  water  ; then  the  correct 

fluent  will  be  s = J6-  X hyp.  log.  of  . 


3.  But  now,  to  determine  v in  terms  of  s,  put  c = 


2*7 1 828 1 828  ; then  since  the  log.  of  — - 
c2bs,  or = c—2bs ; hence  v 


— 2 bs,  therefore 


-7)2 


V a— ac_2bs  is  the  velocity  sought. 
Vol.  II.  - S3 


4.  The 


410 


MOTION  OF  BODIES  IN  FLUIDS. 


4.  The  greatest  velocity  is  to  be  found,  as  in  art.  5 of 


prob.  20,  by  making/  or  1 — 


n 

N 


3r.v2 

16gsd 


0,  which  gives  i 


—ft 

v = y/  (2 g . 8 d . — — — ^/  a.  The  same  value  of  v is 

obtained  by  making  the  fluxion  of  v2 , or  of  a — ac~ 2bs,  = 0. 
And  the  same  value  of  v is  also  obtained  by  making  s in- 
finite, for  then  c-ibs  = 0.  But  this  velocity  y/  a cannot  be 
attained  in  any  finite  time,  and  it  only  denotes  the  velocity  I 
to  which  the  general  value  of  v or  a — ac~  ^ continually 
approaches.  It  is  evident  however,  that  it  will  approximate 
towards  it  the  faster,  the  greater  b is,  or  the  less  d is  ; and  : 
that  the  diameters  being  very  small,  the  bodies  descend  by  i 
nearly  uniform  velocities,  which  are  direct  in  the  subduplicate 
ratio  of  the  diameters.  See  also  art.  5,  prob.  20,  for  other  ob*  r 
servations  on  this  head. 


5.  To  find  the  time  t.  Now  t — - = </  — X — -== 

* a ^/i-c-21* 


Then,  to  find  the  fluent  of  this  fluxion,  put  z — 1 — c~2b»  i 


= — , or  z2  = 1 — c“2b* ; hence  zz  — b's  c~2bs,  and  s =— ■ * . - 
a be — 2» 

1 Zz  , consequently  i — 1 


b l-z2' 


b</a  1 — s2’ 

1 1 + z 

and  therefore  the  fluent  is  t — — — X log.  — = 

' a 8 l-z 


2 b*/c 


2b</a 


X log. 


1 + */  1 - r-*b> 


1 — y'  1 - C-2>>S 

the  general  expression  for  the  time. 


— - — X log.  Ya~^v  which  is 
2 b^a  8 V a — v* 


Exam.  If  it  were  required  to  determine  the  time  and 
velocity,  by  descending  in  air  1000  feet,  the  ball  being  of 
lead,  and  1 inch  diameter. 


Here  n 


Hence  a = 


1 li,  n = d = TV,  and  s — 1000. 

2 . 16tV  • f2«  • ffi  2 . 193 . 8 . 34  2500 


•i 4__ 

• O ^ ft  ft 


3 

193  ' 34  jjg!,  and  ft  J-lInfc- 
9.27  ’ 8.11i.Ti 


3.3  12  . 12  . 3 

3.3.3.12  9.9 


8 . 34  . zoUO  bo  o03’ 


> . <4  sn2 

consequently  v — a X y/  1 — c-2bs  — y/  — — - — — — X 


9-27 

1 


.y/  (1  — c85)  = 203|  the  velocity.  And  t — i X log. 

2 + 


MOTION  OF  BODIES  IN  FLUIDS. 


4ii 


-fy/I— t~2bl  _ 34 . 2500 

1 - v/1— ~ ^ 27  ■ 193 
the  time. 


X 


. 1-78383 

log. 

° 0-21617 


= 8-5236' 


Note.  If  the  globe  be  so  light  as  to  ascend  in  the  fluid  ; it 
is  only  necessary  to  change  the  signs  of  the  first  two  terms 
in  the  value  of  f,  or  the  accelerating  force,  by  which  it  be- 
comes f = — — • 1 — ^'LV  , ; and  then  proceeding  in  all  res- 

•'  N 16gNll  ° 

pects  as  before. 

SCHOLIUM. 


To  compare  this  theory,  contained  in  the  last  four  prob- 
lems, with  experiment,  the  few  following  numbers  are  here 
"Extracted  from  extensive  tables  of  velocities  and  resistances, 
resulting  from  a course  of  many  hundred  very  accurate  ex- 
periments, made  in  the  course  of  the  year  1786. 

F 

; In  the  first  column  are  contained  the  mean  uniform  or 
greatest  velocities  acquired  in  air,  by  globes,  hemispheres, 
cylinders,  and  cones,  all  of  the  same  diameter,  and  the  alti- 
tude of  the  cone  nearly  equal  to  the  diameter  also  when 
urged  by  the  several  weights  expressed  in  avoirdupois 
ounces,  and  standing  on  the  same  line  with  the  velocities, 
aach  in  their  proper  column  So,  in  the  first  line,  the 
numbers  show,  that  when  the  greatest  or  uniform  velocity 
Was  accurately  3 feet  per  second,  the  bodies  were  urged  by 
;hese  weights,  according  as  their  different  ends  went  fore- 
nost ; namely,  by  -028  oz.  when  the  vertex  of  the  cone 
vent  foremost  ; by  -064  oz  when  the  base  of  the  cone  went 
Premost  ; by  -027  oz  for  a whole  sphere  ; by  -050  oz  for 
i cylinder  ; by  -051  oz.  for  the  flat  side  of  the  hemisphere  ; 
and  by  -020  oz.  for  the  round  or  convex  side  of  the  hemis- 
phere. Also  at  the  bottom  of  all,  are  placed  the  mean 
proportions  of  the  resistances  of  these  figures  in  the  nearest 
whole  numbers.  Note,  the  common  diameter  of  ail  the 
figures,  was  6'375,  or  6|  inches  ; so  that  the  area  of  the 
'circle  of  that  diameter  is  jus;  32  square  inches  or  | of  a 
square  foot ; and  the  altitude  of  the  cone  was  6f  inches. 
Also,  the  diameter  of  the  small  hemispheie  was  4|  inches, 
ind  consequently  the  area  of  its  base  17f  square  inches, 
or  i of  a square  foot  nearly. 

From  the  given  dimensions  of  the  cone,  it  appears,  that 
he  angle  made  by  its  side  and  axis,  or  direction  of  the  path, 
s 2^  degrees,  very  nearly. 


The 


412 


MOTION  OF  BODIES  IN  FLUIDS. 


The  mean  height  of  the  barometer  at  the  times  of  making 
the  experiments,  was  nearly  30  1 inches,  and  of  the  ther- 
mometer 62°  ; consequently  the  weight  of  a cubic  foot  of  air 
was  equal  to  1|  oz.  nearly  in  those  circumstances. 


From  this  table  of  resistances,  several  practical  inferences 
may  be  drawn  As, 

1.  That  the  resistance  is  nearly  as  the  surface  ; the  resist- 
ance increasing  but  a very  little  above  that  proportion  in 
the  greater  surfaces.  Thus,  by  comparing  together  the  num- 
bers in  the  6th  and  last  columns,  for  the  bases,  of  the  two- 
hemispheres,  the  areas  of  which  are  in  the  proportion  of 
17f  to  32,  or  as  5 to  9 very  nearly  ; it  appears  that  the 
numbers  in  those  two  columns,  expressing  the  resistances, 
are  nearly  as  1 to  2.  or  as  5 to  10,  as  far  as  to  the  velocity 
of  12  feet  ; after  which  the  resistances  on  the  greater  sur- 
face increase  gradually  more  and  more  above  that  propor- 
tion. And  the  mean  resistances  are  as  140  to  288,  or  as  £ 

to 


Veloc. 

persec. 

Cone. 

Whole 

globe 

Cylin- 

der. 

Hemisphere. 

Small 

Hemis 

flat- 

vertex. 

base 

flat. 

round 

feet. 

oz. 

oz. 

oz. 

oz. 

oz. 

oz. 

oz 

3 

•028 

•064 

•027 

•050 

051 

•020 

•028 

4 

•048 

•109 

•047 

•090 

•096 

•039 

•048 

5 

•071 

•162 

•068 

•143 

•148 

•063 

•072 

6 

•098 

•225 

•094 

•205 

•211 

•092 

•103 

7 

•129 

•298 

•125 

•278 

•284 

•123 

•141 

8 

•168 

•382 

162 

•360 

•368 

•160 

•184 

9 

•211 

•478 

•205 

•456 

•464 

•199 

•233 

10 

•260 

•587 

•255 

•565 

•573 

•242 

•287 

11 

•315 

•712 

•310 

•688 

•698 

•297 

•349 

12 

•376 

•850 

•370 

■826 

•836 

•347 

•418 

13 

•440 

1-000 

•435 

•979 

•988 

•409 

•492 

14 

•512 

1166 

•505 

1145 

1 154 

•478 

•573 

15 

•589 

1 -346 

•581 

1-327 

1-336 

•552 

•661 

16 

•673 

1-546 

•663 

1 526 

1-538 

•634 

•754 

17 

•762 

1-763 

•752 

1-745 

1 -757 

•722 

•853 

18 

•858 

2 002 

•848 

1 986 

1-998 

•818 

•959 

19 

•959 

2-260 

•949 

2-246 

2-258 

•922 

1-073 

20 

1 069 

2-540 

1057 

2 528 

2 542 

1-033 

1196 

Propor 

Numb. 

| 126 

291 

124 

285 

288 

119 

140 

MOTION  OF  BODIES  IN  FLUIDS. 


413 


to  lOf.  This  circumstance  therefore  agrees  nearly  with  the 
theory. 

2.  The  resistance  to  the  same  surface,  is  nearly  as  the 
square  of  the  velocity  ; but  gradually  increasing  more  and 
more  above  that  proportion,  as  the  velocity  increases.  This 
is  manifest  from  all  the  columns.  And  therefore  this  circum- 
stance also  differs  but  little  from  the  theory,  in  small  veloci- 
ties. 

3.  When  the  hinder  parts  ef  bodies  are  of  different  forms, 
the  resistances  are  different,  though  the  fore  parts  be  alike  ; 
owing  to  the  different  pressures  of  the  air  on  the  hinder  parts. 
Thus,  the  resistance  to  the  fore  part  of  the  cylinder,  is  less 
than  that  on  the  flat  base  of  the  hemisphere,  or  of  the  cone  ; 
because  the  hinder  part  of  the  cylinder  is  more  pressed  or 

i pushed,  by  the  following  air,  than  those  of  the  other  two 
figures. 

,4.  The  resistance  on  the  base  of  the  hemisphere,  is  to  that  on 
the  convex  side  nearly  as  2f  to  1,  instead  of  2 to  1,  as  the 
theory  assigns  the  proportion.  And  the  experimented  resist- 
ance, in  each  of  these,  is  nearly  a part  more  than  that  which 
is  assigned  by  the  theory. 

5.  The  resistance  on  the  base  of  the  cone  is  to  that  on  the 
vertex,  nearly  as  2T37  to  1.  And  in  the  same  ratio  is  ra- 
dius to  the  sine  of  the  angle  of  the  inclination  of  the  side  of 
the  cone,  to  its  path  or  axis.  So  that,  in  this  instance,  the 
resistance  is  directly  as  the  sine  of  the  angle  of  incidence,  the 
transverse  section  being  the  same,  instead  of  the  square  of 
the  sine. 

6.  Hence  we  can  find  the  altitude  of  a column  of  air  whose 
pressure  shall  be  equal  to  the  resistance  of  a body,  moving 
through  it  with  any  velocity.  Thus, 

Let  a = the  area  of  the  section  of  the  body,  similar  to 
any  of  those  in  the  table,  perpendicular  to  the 
direction  of  motion  ; 

r = the  resistance  to  the  velocity,  in  the  table  ; and 
x — the  altitude  sought,  of  a column  of  air,  whose 
base  is  ct,  and  its  pressure  r. 

Then  ax  = the  content  of  the  column  in  feet, 
and  liax  or  \ax  its  weight  in  ounces  ; - 

therefore  fax  — ?•,  and  x = £ X — is  the  altitude  sought  in 

feet, 


414 


MOTION  OF  BODIES  IN  FLUIDS. 


feet,  namely,  f of  the  quotient  of  the  resistance  of  any  body 
divided  by  its  transverse  section  ; which  is  a constant  quan- 
tity for  all  similar  bodies,  however  different  in  magnitude, 
since  the  resistance  r is  as  the  section  a,  as  was  found  in  art.  1. 
When  a — f of  a foot,  as  in  all  the  figures  in  the  forego- 
ing table,  except  the  small  hemisphere  : then,  x — J-  X — 

< a 

becomes  x = \5r,  where  r is  the  resistance  in  the  table,  to 
the  similar  body . 

If,  for  example,  we  take  the  convex  side  of  the  large  hemis- 
phere, whose  resistance  is  -634  oz.  to  a velocity  of  16  feet  per 
second,  then  r = 634,  and  i = {r  = 2‘3775  feet,  is  the  alti- 
tude of  the  column  of  air  whose  pressure  is  equal  to  the  resist- 
ance on  a spherical  surface,  with  a velocity  of  16  feet.  And 
to  compare  the  above  altitude  with  that  which  is  due  to  the 
given  velocity,  it  will  be  322  : 162  : : 16  : 4,  the  altitude  due 
to  the  velocity  16  ; which  is  near  double  the  altitude  that  is 
equal  to  the  pressure.  And  as  the  altitude  is  proportional  to 
the  square  of  the  velocity,  therefore,  in  small  velocities,  the 
resistance  to  any  spherical  surface  is  equal  to  the  pressure  of 
a column  of  air  on  its  great  circle,  whose  altitude  is  if  or  -594 
of  the  altitude  due  to  its  velocity. 

But  if  the  cylinder  be  taken,  whose  resistance  r = 1-526  : 
then  x — L®  r = b-12  ; which  exceeds  the  height,  4,  due 
to  the  velocity  in  the  ratio  of  23  to  16  nearly.  And  the  differ- 
ence would  be  still  greater,  if  the  body  were  larger  ; and  also 
if  the  velocity  were  more. 

7.  Also,  if  it  be  required  to  find  with  what  velocity  any  flat 
surface  must  be  moved,  so  as  to  suffer  a resistance  just  equal 
to  the  whole  pressure  of  the  atmosphere  : 

The  resistance  on  the  whole  circle  whose  area  is  f of  a foot, 
is  051  oz.  with  the  velocity  of  3 feet  per  second  ; it  is  i of 
•051,  or  0056  oz.  only,  with  a velocity  of  1 foot.  But  2^-  X 
13600  X | = 7555f-  oz.  is  the  whole  pressure  of  the  atmos- 
phere. Therefore,  as  0056  : 7556  : : 1 : 1162  nearly, 

which  is  the  velocity  sought.  Being  almost  equal  to  the  velo- 
city with  which  air  rushes  into  a vacuum. 

8 Hence  may  be  inferred  the  great  resistance  suffered  by- 
military  projectiles.  For  in  the  table,  it  appears,  that  a globe 
of  6f  inches  diameter  which  is  equal  to  the  size  of  an  iron  ball 
weighing  36lb,  moving  with  a velocity  of  only  16  feet  per  se-. 
cond,  meets  with  a resistance  equal  to  the  pressure  of  f of  an 
ounce  weight ; aad  therefore,  computing  only  according  to  the 

square 


MOTION  OF  BODIES  IN  FLUIDS. 


415 


square  of  the  velocity,  the  least  resistance  that  such  a ball 
would  meet  with,  when  moving  with  a velocity  of  16G0  feet 
would  be  equal  to  the  pressure  of  417  lb,  and  that  independent 
of  the  pressure  of  the  atmosphere  itself  on  the  fore  part  of  the 
bell  which  would  be  4871b  more,  as  there  would  be  no  pres- 
sure from  tbe  atmosphere  on  the  hinder  part,  in  the  case  of  so 
great  a velocity  as  1600  feet  per  second.  So  that  the  whole 
I resistance  w'ould  be  more  than  900lb  to  such  a velocity. 

9.  Having  said,  in  the  last  article,  that  the  pressure  of  the 
atmosphere  is  taken  entirely  off  the  binder  part  of  the  ball 
moving  with  a velocity  of  1600  feet  per  second  ; which  must 
happen  when  the  ball  moves  faster  than  the  particles  of  air 

. can  follow  by  rushing  into  the  place  quitted  and  left  void  by 
the  ball,  or  when  the  ball  moves  faster  than  the  air  rushes 
into  a vacuum  from  the  pressure  of  the  incumbent  air  ; let 
us  therefore  inquire  what  this  velocity  is.  Now  the  velocity 
with  which  any  fluid  issues,  depends  on  its  altitude  above 
the  orifice,  and  is  indeed  equal  to  the  velocity  acquired  by 
a heavy  body  in  falling  freely  through  that  altitude.  But, 
supposing  the  height  of  the  barometer  to  be  30  inches,  or 

feet,  the  height  of  a uniform  atmosphere,  all  of  the  same 
density  as  at  the  earth’s  surface,  would  be  2|  X 13-6  X 833  a 
or  28333  feet ; therefore  16  : y/  28333  : : 32  : 8 28333 

= 1346  feet,  which  is  the  velocity  sought  And  therefore, 
with  a velocity  of  1600  feet  per  second,  or  any  velocity 
above  1346  feet,  the  ball  must  continually  leave  a vacuum 
behind  it,  and  so  must  sustain  the  whole  pressure  of  the  at- 
mosphere on  its  fore  part,  as  well  as  the  resistance  arising 
from  the  vis  inertia  of  the  particles  of  air  struck  by  the  ball. 

10.  On  the  whole,  we  find  that  the  resistance  of  the  air,  as 
determined  by  the  experiments,  differs  very  widely,  both  in 
respect  to  its  quantity  on  all  figures,  and  in  respect  to  the  pro- 
portions of  it  on  oblique  surfaces,  from  the  same  as  determin- 
ed by  the  preceding  theory  ; which  is  the  same  as  that  of  Sir 
Isaac  Newton,  ?ud  most  modern  philosophers.  Neither  should 
we  succeed  better  if  we  have  recourse  to  the  theory  given  by 
Professor  Gravesande,  or  others,  as  similar  differences  and  in- 
consistencies still  occur. 

We  conclude  therefore,  that  all  the  theories  of  the  resist- 
ance of  the  air  hitherto  given  are  very  erroneous.  And  the 
preceding  one  is  only  laid  clown,  till  further  experiments,  on 
this  important  subject,  shall  enable  us  to  deduce  from  them 
another,  that  shall  be  more  consonant  to  the  true  phenomena 
of  nature. 


ON 


L 416  ] 


ON  THE  MOTION  OF  MACHINES,  AND  THEIR 
MAXIMUM  EFFECTS. 


Art.  1.  When  forces  acting  in  contrary  directions,  or 
in  any  such  directions  as  produce  contrary  effects,  are  ap- 
plied to  machines,  there  is,  with  respect  to  every  simple  ma- 
chine (and  of  consequence  with  respect  to  every  combination 
of  simple  machines)  a certain  relation  between  the  powers 
and  the  distances  at  which  they  act,  which,  if  subsisting  in 
any  such  machine  when  at  rest,  will  always  keep  it  in  a state 
of  rest,  or  of  statical  equilibrium  ; and  for  this  reason,  be- 
cause the  efforts  of  these  powers  when  thus  related,  with  t 
regard  to  magnitude  and  distance,  being  equal  and  opposite 
annihilate  each  other,  and  have  no  tendency  to  change  the 
state  of  the  system  to  which  they  are  applied.  So  also,  if 
the  same  machine  Have  been  put  into  a state  of  uniform,  mo- 
tion, whether  rectilinear  or  rotatory,  by  the  action  of  any 
power  distinct  from  those  we  are  now  considering,  and  these 
two  powers  be  made  to  act  upon  the  machine  in  such  motion 
in  a similar  manner  to  that  in  which  they  acted  upon  it  when 
at  rest,  their  simultaneous  action  will  preserve  it  in  that  state 
of  uniform  motion,  or  of  dynamical  equilibrium  : and  this  for 
the  same  reason  as  before,  because  their  contrary  effects  de- 
stroy each  other,  and  have  therefore  no  tendency  to  change 
the  state  of  the  machine.  But,  if  at  the  time  a machine  is 
in  a state  of  balanced  rest,  any  one  of  the  opposite  forces  be 
increased  wThile  it  continues  to  act  at  the  same  distance,  this 
excess  of  force  will  disturb  the  statical  equilibrium,  and  pro- 
duce motion  in  the  machine  ; and  if  the  same  excess  of  force 
continues  to  act  in  the  same  manner,  it  will,  like  every  con- 
stant force,  produce  an  accelerated  motion  ; or  if  it  should 
undergo  particular  modifications  when  the  machine  is  in  dif- 
ferent positions,  it  may  occasion  such  variations  in  the  motion 
as  will  render  it  alternately  accelerated  and  retarded  Or  the 
different  species  of  resistance  to  which  a moving  machine  is 
subjected,  as  the  rigidity  of  ropes,  friction,  resistance  of  the 
air,  &c.  may  so  modify  a motion,  as  to  change  a regular  or  ir- 
regular variable  motion  into  one  which  is  uniform. 

2 Hence  then  the  motion  of  machines  may  be  considered 
as  of  three  kinds.  1.  That  which  is  gradually  accelerated, 
which  obtains  commonly  in  the  first  instants  of  the  commu- 
nication 2.  That  which  is  entirely  uniform.  3.  That  which 
is  alternately  accelerated  and  retarded.  Pendulum  clocks, 
and  machines  which  are  moved  by  a balance,  are  related  to 


MAXIMUM  IN  MACHINES. 


417 


che  third  class.  Most  other  machines,  a short  time  after 
their  motion  is  commenced,  fall  under  the  second.  Now 
though  the  motion  of  a machine  is  alternately  accelerated 
and  retarded,  it  may,  notwithstanding,  be  measured  by  a 
uniform  motion,  because  of  the  periodical  and  regular  repe- 
tition which  may  exist  in  the  acceleration  and  retardation. 
Thus  the  motion  of  a second’s  pendulum,  considered  in  re- 
spect to  a single  oscillation,  is  accelerated  during  the  first  half 
second,  and  retarded  during  the  next  : but  the  same  motion 
taken  for  many  oscillations  may  be  considered  as  uniform. 
Suppose,  for  example,  that  the  extent  of  each  oscillation  is 
5 inches,  and  that  the  pendulum  has  made  10  oscillations  : 
its  total  effect  will  be  to  have  run  over  50  inches  in  10  se- 
conds ; and,  as  the  space  described  in  each  second  is  the 
same,  we  may  compare  the  effect  to  that  produced  by  a 
moveable  which  moves  for  10  seconds  with  a velocity  of  5 
inches  per  second.  We  see,  therefore,  that  the  theory  of 
machines  whose  motions  are  uniform,  conduces  naturally  to 
the  estimation  of  the  effects  produced  by  machines  whose 
motion  is  alternately  accelerated  and  retarded  : so  that  the 
problems  comprised  in  this  chapter  will  be  directed  to  those 
machines  whose  motions  fall  under  the  first  two  heads  ; such 
problems  being  of  far  the  greatest  utility  in  practice. 

Defs.  1.  When  in  a machine  there  is  a system  of  forces 
or  of  powers  mutually  in  opposition,  those  which  produce  or 
tend  to  produce  a certain  effect  are  called,  movers  or  moving 
powers  ; and  those  which  produce  or  tend  to  produce  an 
effect  which  opposes  those  of  the  moving  powers,  are  called 
resistances,  if  various  movers  act  at  the  same  time,  their 
equivalent  (found  by  means  of  prob.  7,  Motion  and  Forces) 
is  called  individually  the  moving  force ; and,  in  like  manner, 
the  resultant  of  all  the  resistances  reduced  to  some  one  point, 
the  resistance.  This  reduction  in  all  cases  simplifies  the  in- 
vestigation. 

2.  The  impelled  point  of  a machine  is  that  to  which  the 
action  of  the  moving  power  may  be  considered  as  immedi- 
ately applied  ; and  the  working  point  is  that  where  the  re- 
sistance arising  from  the  work  to  be  performed  immediately 
acts,  or  to  which  it  ought  all  to  be  reduced.  Thus,  in  the 
wheel  and  axle,  (Meehan,  prop  32),  where  the  moving 
power  p is  to  overcome  the  weight  or  resistance  w,  by  the 
application  of  the  cords  to  the  wheel  and  to  the  axle,  b is  the 
• impelled  point,  and  a the  working  point. 

54 


Vol.  II. 


3.  The 


4i8 


MAXIMUM  IN  MACHINES. 


3.  The  velocity  of  the  moving  power  is  the  same  as  the 
velocity  of  the  impelled  point  ; the  velocity  of  the  resistance 
the  same  as  that  of  the  working  point. 

4.  The  performance  or  effect  of  a machine,  or  the  work . 
done , is  measured  by  the  product  of  the  resistance  into  the 
velocity  of  the  working  point ; the  momentum  of  impulse  is 
measured  by  the  product  of  the  moving  force  into  the  velo- 
city of  the  impelled  point. 

These  definitions  being  established  we  may  now  exhibit  a 
* few  of  the  most  useful  problems,  giving  as  much  variety  in 
their  solutions  as  may  render  one  or  other  of  the  methods  of 
easy  application  to  any  other  cases  which  may  occur. 

PROPOSITION  I, 


If  r,  and  r be  the  distances  of  the  power  p,  and  the  weight  or 
resistance  w,  from  the  fulcrum  f of  a straight  lever  : then  will 
the  velocity  of  the  power  and  of  the  weight  at  the  end  of 


any  time  t be 


R P — Rcw 


gt,  and 


R p-f2\V 


gt,  respectively,  the 


R-’p+r2w6  R!p-)-i!w 

weight  and  inertia  of  the  lever  itself  not  being  considered. 


If  the  effort  of  the  power  ba- 
lanced that  of  the  resistance,  p 

would  be  equal  to—.  Conse- 


A 

yP 


A 


W 


quently,  the  difference  between  this  value  of  p,  and  its  actual 
value,  or  p — — w,  will  be  the  force  which  tends  to  move 

K 

the  lever.  And  because  this  power  applied  to  the  point  a 
accelerates  the  masses  r and  w,  the  mass  to  be  substituted 

for  w,  in  the  point  a,  must  be  r-,  (Meehan,  prop.  50)  in 

order  that  this  mass  at  the  distance  r may  be  equally  accele- 
rated with  the  mass  w at  the  distance  r.  Hence  the  power 

p — w will  accelerate  the  quantity  of  matter  p + — w ; and 

the  accelerating  force  f = (p  — -w)-j-(p+  r— w)=  F~ — 

& V r J V R2  ' PR2+r2w 

But(  Art33,Gen.LawsofMotion)va  Ftoris  =gtF(igbeing=‘=o2^ 
feet);  which  in  this  case  = R P R ^ .gt,  the  velocity  of  p And, 

R2  P + .3W  ° J 

because  veloc  of  p : veloc  of  w : : r : r,  therefore  veloc.  of 


w = — veloc.  of  p ==  — 
R R 


RaP— R w RrP  — r2  w 

X gt  = -St. 


R 2P-f-r2  w ' 


R2  p-pr2  w 


Corol. 


41& 


MAXIMUM  IN  MACHINES. 

. I 

Corol.  1.  The  space  described  by  the  power  in  the  time  t, 

I«  R **  P ««  • 

will  be  = — . ±gt2  ; the  space  described  by  w in  the 

R2P  + r2W  20  ’ r J 

,.  -ii  l KrP  — r' w , 

same  time  will  be  — — . igt2. 

R3F  + R2W  20 

Cor.  2.  If  r : r ; : n : l,then  will  the  force  which  acce- 

, , Vn2  — w?z 

lerates  a be  = . 

F/!2  -f  - w 

Cor.  3.  If  at  the  same  time  the  inertia  of  the  moving 
force,  p be  = 0,  as  in  muscular  action,  the  force  accelerating 

■I,,  P/12—W-1 

a will  be  = . 

w 

Cor  4.  If  the  mass  moved  have  no  weight,  but  possesses 
tnertia  only,  as  when  a body  is  moved  along  a horizontal 

P * 

plane,  the  force  which  accelerates  a will  be  = . And 

P>i2-pW 

either  of  these  values  may  be  readily  introduced  into  the  in- 
vestigation. 

II  ' 

Cor.  5.  The  work  done  in  the  time  t,  if  we  retain  the  ori- 
ginal  notation,  will  be  = gt  X w = . gi. 

° R2p+r2W°  R2P^.i-2w  6 

Cor.  6.  When  the  work  done  is -to  be  a maximum,  and  we 
wish  to  know  the  weight  when  p is  given,  we  must  make 
the  fluxion  of  the  last  expression  = 0.  TheD  we  shall  have, 

i-R3P2 — 2r2R2pw  — r4w 2=  0 and  w = pX[V(R — |-R-  ) — — 

■- v \ ,,4  r^j  r2  J 

Cor.  7.  If  r : r : : n : 1,  the  preceding  expression  will  be- 
comew  = p X [v/(w44*^3)  — n2]. 

Cor.  8.  When  the  arms  of  the  lever  are  equal  in  length, 
that  is,,  when  n = 1,  then  is  w = p X (^2— 1)  = -414214P. 
or  nearly  Ts2  of  the  moving  force. 

Scholium. 


If  we  in  like  manner  investigate  the  formulas  relating  to 
motion  on  the  axis  in  peritrochio,  it  will  be  seen  that  the 
expressions  correspond  exactly.  Hence  it  follows,  that  when 
it  is  required  to  proportion  the  power  and  weight  so  as  to 

obtain 


420 


MAXIMUM  IN  MACHINES. 


obtain  a maximum  effect  on  the  wheel  and  axle,  (the  weight 
ot  the  machinery  not  being  considered),  we  may  adopt  the 
conclusions  of  cors.  6 and  7 of  this  prop.  And  in  the'  extreme 
case  where  the  wheel  and  axle  becomes  a pulley,  the  expres- 
sion in  cor.  8 may  be  adopted.  The  like  conclusions  .may  be 
applied  to  machines  in  general,  if  r and  r represent  the  dis- 
tances of  the  impelled  and  working  points  from  the  axis  of 
molion  ; and  if  the  various  kinds  of  resistance  arising  from 
friction,  stiffness  of  ropes,  &c.  be  properly  reduced  to  their 
equivalents  at  the  working  points,  so  as  to  be  comprehended 
in  the  character  w for  resistance  overcome  . 


PROPOSITION  II. 

Given  r and  r,  the  arms  of  a straight  lever,  m and  m theii 
respective  weights,  and  p the  power  acting  at  the  extremity  of 
the  arm  r ; to  find  the  weight  raised  at  the  extremity  of -the  other 
arm  when  the  effect  is  a maximum. 

In  this  case  is  the  weight  of 
the  shorter  end  reduced  to  b,  and 

conseq.  0—  is  the  weight  which 

applied  at  a,  would  balance  the 

— + -w,  would  sustain  both  the  shorter  end  and  the  weight 
2r  r 

w in  equilibrio.  But  p+i  m *s  Power  really  acting  at  the 
longer  end  of  the  lever  ; consequently 

p + isi  — (— w)>  is  the  absolute  moving  power.  Now 
the  distance  of  the  centre  of  gyration  of  the  beam  from  f*- 


T 

~tv~ 


shorter  end  ; therefore 


* The  distance  of  R,  the  centre  of  gyration,  from  c the  centre  or 
axis  of  motion,  in  some  of  the  most  useful  cases,  is  as  below  ; 


In  a circular  wheel  of  uniform  thickness  ....  cr  = rad.^/4 
In  the  periphery  of  a circle  revolving  about  the  dlam."  CR  = rad. ^4 
In  the  plane  of.  a circle  . . . ditto  . . . cr  = 4 r^d-  „ 

In  the  suiface  of  a sphere  . . . ditto  ...  cr  = rad.^/f 

In  a solid  sphere  . . ....  ditto  . . . cr  *=  rad-v'f 

In  a plane  ring  formed  of  circles  whose  radii  are  7 R* 

r,  r,  revolving  ahout  centre y CR==v,2k2 2i-‘ 


In  a ci  ne  revolving  about  Its  vertex  . . . . ■ 

In  a cone its  axis 

in  a straight  lever  whose  arms  arc  r and  r . 


CR  = >V  r\ 

3-j-r3 

cf.=^/  — .. 

(oR  +r) 


MAXIMUM  IN  MACHINES. 


- ' 425 


is  = jy/  , which  let  be  denoted  by  f ; then  (Meehan. 

prop.  50)  — , (m  ,-f-  m)  will  represent  the  mass  equivalent 

to  the  beam  or  lever  when  reduced  to  the  point  a ; while 
the  weight  equivalent  to  w,  when  referred  to  that  point,- 

i’5  b - 

will  be  — w.  Hence,  proceeding  as  in  the  last  prop,  we 

p 2 p2> 

shall  have  — , (m  + + p wforthe  inertia  to  beover- 

. R-  ' ' R2 

pjp  >p  p 2,  p2 

come  ; and  (p  + Jm— — — -w)  ^ (m  + m)  + p + — 


R2 


= the  accelerating  force  of  f,  or  o^  w reduced  to  a.  Mul- 
tiply this  by  w ; and,  for  the  sake  of  simplifying  the  pro- 
cess, put  q for  p -}-  isi  — ~ , and  n for  p 4 (m  + m). 


q\v  ■ 


r w3 

R 


then  will 


w 

R3 


be  a quantity  which  varies  as  the  effect 


varies,  and  which,  indeed,  when  multiplied  by  gt,  denotes 
the  effect  itself.  Putting  the  fluxion  of  this  equal  to  nothing, 
I and  reducing,  we  at  length  find 


R .nqR 


n-  R- 


nr2 

■7T- 


Cor.  When  r = r,  and  m = m,  if  we  restore  the  values 
i of  n and  q , the  expression  will  become  w = (2p2  4 2wir 

~f  fm2)  — (p  4-  fm). 

PROPOSITION  III. 

j .Given  the  length  l and  angle  e of  elevation  of  an  inclined 
'plane  bc  ; to  find  the  length  i.  of  another  inclined  plane  ac  along 
I which  a given  weight  w shall  be  raised  from  the  horizontal  line 
! ab  to  the  point  c,  in  the  least  time  possible,  by  means  of  another 
i given  weight  p descending  along  the  given  plane  cb  : the  two 
i weights  being  connected  by  an  inextensible  thread  bcW  running 
always  parallel  to  the  two  planes. 

Here  we  must,  as  a preliminary 
to  the  solution  of  this  proposition, 
deduce  expressions  for  the  motion 
of  bodies  connected  by  a thread,  and 
running  upon  double  inclined  planes. 

Let  the  angle  of  elevation  cad  be 
e,  while  e is  the  elevation  cbd. 

• Then  at  the  end  of  the  time  t,  p 


will 


422  • 


MAXIMUM  IN  MACHINES. 


will  have  a velocity  v ; and  gravity  would  impress  upon  it 
in  the  instant  \ following,  a new  velocity  = g sin  e . / , pro- 
vided the  weight  p were  then  entirely  free  : but,  by  the  dis- 
position of  the  system,  v will  be  the  velocity  which  obtains 
in  reality.  Then,  estimating  the  spaces  in  the  direction  cp, 
as  the  body  w moves  with  an  equal  velocity  but  in  a contrary 
sense,  it  is  obvious,  that  by  applying  the  3d  Law  of  Motion, 
the  decomposition  may  be  made  as  follows.  At  the  end  of  the 
time  t + t we  have,  for  the  velocity  impressed  on, 

, _ . , Cjy  4-i,  • • • • • effective  vetcc.from  c towards  B- 

1 * ’ \t*  sin  e . t — t>, . . . . velocity  cest  oyed. 

. . . , S —v— l ..  effective  veloc.fr  m c towards  a. 

w.-v  + g sm  E.  t,  where  sin  E . ; . . velocity  destroyed. 

If,  therefore,  gravity  impresses,  during  the  time  j upon  the 
masses  p,  w,  the  respective  velocities  g sin  c . t — and  g 
sin  e . ; + v,  the  system  will  be  in  equihbrio.  The  quanti- 
ties of  motion  being  therefore  equal,  it  will  be 

pg  sin  e . } — Pv  — w g sin  e . i 4"  w-^. 

Whence  the  effective  accelerating  force  is  found,  i.  e. 

. v.  T sin  e — w sin  e 

d>  = —= X g. 

Thus  it  appears  that  the  motion  is  uniformly  varied,  and  we 
readily  find  the  equations  for  the  velocity  and  space  from . 
which  the  conditions  of  the  motion  are  determined  : viz. 


p sin  e—  w sin  e 


p sin  e — w s.n  e 


W*- 

i(p  + w) 


p -J-  W p T W 

The  latter  of  these  two  equations  gives  t2=  ■ 

psinc-wj  :v) 

But  in  the  triangle  abc  it  is  ac  : bc  : : sin  b : sin  a,  that  is, 

h : l : : sin  e : sin  e ; hence  — l = sin  e,  and  — l = sin  e ; 

m m 

■ m being  a constant  quantity  always  determinable  from  the 

data  given.  And  i2  becomes  — — - — . Now  when  any 

2(PI.  - wZ) 
as  m y 

quantity,  as  t,  is  a minimum,  its  square  is  manifestly  a mini- 
mum : so  that  substituting  for  s its  equal  l,  and  striking  out 

L2 

the  constant  factors,  we  have = a min.  or  its  fiuxion 


2ll  ( PL  — wl— PL2 e 


=0. 


rL  — wl 

Herq,  as  in  all  similar  cases,  since 


(p  L — wt) 

the  fraction  vanishes,  its  numerator  must  be  equal  to  0 ; con- 
sequently 2pl2  — 2 w/l  - pl2  =0,  pl  ==  2\vZ,  or  l : / : : 
2w  : p. 

Cor.  1.  Since  neither  sin  e nor  sin  e enters  the  final  equa- 
tion, it  follows,  that  if  the  elevation  of  the  plane  bc  is  not 
given,  the  problem  is  unlimited 

Cor. 


MAXIMUM  IN  MACHINES. 


423 


Cor.  2.  When  sin  e = 1,  bc  coincides  with  the  perpen- 
dicular cd,  and  the  power  p acts  with  all  its  intensity  upon  the 
weight  w.  This  is  the  case  of  the  present  problem  which 
has  commonly  been  considered. 

Scholium. 

This  proposition  admits  of  a neat 
geometrical  demonstration.  Thus, 
let  ce  be  the  plane  upon  which,  if 
w were  placed,  it  would  be  sus- 
tained in  equilibrio  by  the  power  p 
on  the  plane  cb,  or  the  power  p 
hanging  freely  in  the  vertical  cd  ; 
then  (Meehan,  prop.  23)  bc  : cd  : ce  : p : p'  : w.  But 
w is  to  the  force  with  which  it  tends  to  descend  along 
the  plane  ca,  as  ca  to  cd  ; consequently,  the  weight  r is  to 
the  same  force  in  the  same  ratio  ; because  either  of  these 
weights  in  their  respective  positions  would  sustain  w on  ce, 
Therefore  the  excess  of  p above  that  force  (which  excess  is 
the  power  accelerating  the  motions  of  p and  vv)  is  to  p,  as 
ca  — ce  to  ca  ; or  taking  ch  = ca,  as  eh  to  ca.  Now, 
the  motion  being  uniformly  accelerated,  we  have  s oc  ft2,  or 

consequently,  the  square  of  the  time  in  which  ac 


a 


is  described  by  w,  will  be 


as  ac  directly,  and  as  — - in- 


a minimum ; that  is. 


versely  ; and  will  be  least  when  is 
. eu 

when  4-  eh  + 2ce,  or  (because  2ce  is  invariable)  when 


ce2  . 

+ eh  is 

EH 


a minimum.  Now,  as,  when  the  sum  of  two 

|uantities  is  given,  their  product  is  a maximum  when  thev 
are  equal  to  each  other  ; so  it  is  manifest  that  when  their 
product  is  given,  their  sum  must  be  a minimum  when  they 

CE  2 

are  equal.  But  the  product  of  — and  eh  is  ce2,  and  con- 


sequently given  ; therefore  the  sum  of  *■——  and  eh  is  least 

E H. 

when  those  parts  are  equal  ; that  is  when  eh  = cf.,  or 
ca  = 2ce.  So  that  the  length  of  the  plane  ca  is  double  the 
length  of  that  on  which  the  weight  w would  be  kept  in  equili- 
brio by  p acting  along  cb. 

When  cd  and  cb  coincide,  the  case  becomes  the  same  as 
that  considered  by  Maclaurin,  in  his  View  of  Newton's  Philo- 
sophical Discoveries,  pa.  183,  8vo.  edit 


FEOPOS1TIOM 


424 


MAXIMUM  IN  MACHINES. 
PROPOSITION  IV. 


Let  the  given  •weight  p descend  along  cb,  and  by -means  of  the 
thread  pcw  ( running  parallel  to  the  planes)  draw  a weight  w 
■up  the  plane  ac  : it  is  required  to  find  the  values  of  w,  when  its 
momentum  is  a maximum , the  lengths  and  positions  of  the  planes 
being  given.  (See  the  preceding  Jig.) 

The  general  expression  for  the  vel  in  v = P”  'n<\_— 

• which,  by  substitut.  — l for  sin  e,  and  — l for  sin  e,  becomes 


— (pl— wZ) 

m v ' . 

V = gt, 


This  mul.  into  w, gives - 


-(PWL-W  2l) 


-st; 


P+W.  P-pW 

which,  by  the  prop,  is  to  be  a maximum.  Or,  striking  out 

1 . PWL— 

the  constant  factors,  — , gt,  then  is =--  a max.  Put- 

771  P-J-W 

ting  this  into  fluxions,  and  reducing,  we  have  p2l — 2pwZ— 

w2l  = 0,orw  = pv/(j  + 1)  — p. 

Cor.  When  the  inclinations  of  the  planes  are  equal,  l,  and 
l are  equal,  and  w = pv/2  — p = p X (^/  2 — 1)  = -4142p  : 
agreeing  with  the  conclusion  of  the  lever  of  equal  arms,  or 
the  extreme  ease  of  the  wheel  and  axle,  i.  e.  the  pulley. 

PROPOSrriON  V. 

Given  the  radius  r of  a wheel,  and  the  radius  r of  its  axle , 
the  weight  of  both,  w,  and  the  distance  of  the  centre  of  gyra- 
tion from  the  axis  of  motion  % ; also  a given  power  p acting  at 
■ the  circumference  of  the  wheel;  to  find  the  weight  w raised 
by  a cord  folding  about  the  axle,  so  that  its  momentum  shall  be  a 
maximum. 

The  force  which  absolutely  impels 
the  point  a is  p,  while  w acts  in  a 
direction  contrary  to  p^  with  a force  — 

— ; this  therefore  subducted  from  p, 
r ’ 

, r\v  RP-rw  . ,, 

leaves  p = — — — , tor  the  re- 

k r 

duced  force  of  impelling  the  point  a. 

And  the  inertia  which  resists  the  com- 
munication of  motion  to  the  point  a will  be  the  same  as  if 
e2  w-4-r2  w+r2  p 

the  mass  — were  concentrated  in  the  pointx  (Me- 

ehan. prob.  50).  If  the  former  of  these  be  divided  by  the 
latter,  the  quotient  — — — - — rm  ■>s  the  force  accelerating  a : 


S2W+r2w-f-R2P 


multiplying 


MAXIMUM  IN  MACHINES. 


425 


~ v RrP  - r 2 iv 

multiplying  this  by  — , we  have  — — for  the  force 

J R 5:w4-f2w-fR2p 

which  accelerates  the  weight  w in  its  ascent.  Consequently 

nr p — i 2 w 

the  velocity  of  w will  be  = ■ o— — —gt  \ which  multi- 


RrPW — r * w 


§2?j)-(-|;lw+R2pC 

gt  for  the  momentum.  As  this 


plied  into  w gives 6. 

r ° W-{-R2F 

is  to  be  a maximum,  its  fluxion  will  = 0 ; whence  we  shall 

, , . ^(R4P2+2R2Pp2w+»4TO2_l.Pj1)Rro2+p3  R3,,)_R2p_e3w 

obtain  vv  = — — ?— 

r2 

Cor.  1.  When  r = r,  as  in  the  case  of  the  single  fixed  pul- 
ley, then  w = v/(2p2r3-1-2rp£2?£>4-^—  «>2-{-p:»r£2)— — ^ w— p. 

Cor.  2.  When  the  pulley  is  a cylinder  of  uniform  matter 
=|rj, and  the  express,  becc  mes  w =x/[R3(2p2+|pzii;^-itiy2)J 

—±W'~p. 

Cor.  3.  If,  in  the  first  general  expression  for  the  mo- 

R(’PW— r2  vv2 

mentum  of  w,  q be  put  = r2f-\-p2w,  we  shall  have 

’ 1 q.  f-  r 2 W 

Which,  in  fluxions  and  reduced,  gives  w = 

1 


= a maximum 
1 


— ^ • (ft  + Rrp)  — Q. 

Cor.  4.  If  the  moving  force  be  destitute  of  inertia,  then 
will  q = and  w,  as  in  the  last  corollary. 


PROPOSITION  VI. 


Let  a given  power  p be  applied  to  the  circumference  of  a wheel, 
its  radius  r,  to  raise  a weight  w at  its  axle,  whose  radius  is 
r,  it  is  required  to  find  the  ratio  of  r and  r when  w is  raised 
with  the  greatestimomentum  ; the  characters  w and  % denoting  the 
same  as  in  the  last  proposition. 

Here  we  suppose  r to  vary  in  the  expression  for  the  mo- 

"WR  7"P  — /’ 2 \y2 

raentum  of  w,  — , gt.  And  we  suppose,  that  by 

the  conditions  of  any  specified  instance,  we  can  ascertain  what 
quantity  of  matter  q shall  make  r^q  = %2w,  which,  in  fact, 
may  always  be  done  as  soon  as  we  c n determine  f.  The  ex- 

Rrp  vv  — r2  w2 

pression  for  the  work  will  then  become  — ; — -gt.  The 

1 R2p+r2(y+w)° 

fluxion  of  which  being  made  = 0,  gives,  alter  a little  reduc- 

Rv/fP2  w2-f  pS(y-(-TO)]  _ pw 

ion,  r = — • 

p(?+  w) 

Cor.  When  the  inertia  of  the  machine  is  evanescent,  with 

p 

espect.to  that  of  £ + w,  then  is  r — r (1  + -)  — 1. 
Vol.  If.  55  • PROPOSITION 


426 


MAXIMUM  IN  MACHINES. 


PROPOSITION  VII. 


In  any  machine  whose  motion  accelerates,  the  weight  will 
be  moved  with  the  greutest  velocity,  when  the  velocity  of  the 


■power  is  to  that  of  the  weight,  as  1 -f-  p ^/(l  H — ) to  1 ; the 


inertia  of  the  machine  being  disregarded 

For  any  such  machine  may  be  considered  as  reduced  to  a 
lever,  or  to  a wheel  and  axle  whose  radii  are  r and  r : in 


R Up  p2  W 

which  the  velocity  of  the  weight  — gt  (prop.  1)  is  to 


be  a maximum,  r being  considered  as  variable.  Hence  then,  1 


following  the  usual  rules,  we  find  pr  = r (w  + y/  w2  + pvt). 
From  which,  since  the  velocities  of  the  power  and  weight  are 
respectively  as  r and  r,  the  ratio  in  the  proposition  immediate- 
ly flows. 

Cor.  When  the  weight  moved  is  equal  to  the  power,  then  is 
r : r : : 1 -f  y'J  ; 1 ::  2 4 142  : 1 nearly. 


proposition  vm. 


to  1J-  and  1 to  2. 


Let  the  space  descended  be  1,  that  ascended s ; the  de- 


scending \Veight  1,  the  ascending  weight  — : then  would  the 


equilibrium  require  w = s ; and  1 will  be  the  force  act- 

ing on  1.  Now  the  mass  — , reduced  to  the  point  at  which 


the  mass  1 acts,  will  be 


i3 


s2  = — ; consequently  the 


whole  mass  moved  is  equivalent  to  1 -) , and  the  relative 

w 


force  is  (1  — — ) -4-  (1  ) = , . But,  the  space  be- 

v -w ' x W W+s2 

rng  given,  the  time  is  as  the  root  of  the  accelerating  force 


^-sL.^2  # 

inversely,  that  is,  as  y/ : and  the  whole  effect  in  a given 


time,  being  directly  as  the  weight  raised,  and  inversely  as 

which  must  be  a 
maximum 


the  time  of  ascent,  will  be  as  — y/  — — - 

TO  TP-f-*2 


If  in  any  machine  whose  motion  accelerates,  the  descent  of 
one  weight  causes  another  to  ascend,  and  the  descending 
zsfeight  be  given,  the  operation  being  supposed  continually  re- 
peated, the  effect  will  be  greatest  in  a given  time  when  the 
ascending  weight  is  to  the  descending  weight,  as  1 to  1-618, 
in  the  case  of  equal  heights  ; and  in  other  cates,  when  it  is  to 
the  exact  counterpoise  in  a ratio  which  is  always  between  1 


MAXIMUM  IN  MACHINES. 


42? 


W— $ 


aminoum.  Consequently  its  square 
likewise.  This  latter  expression, 
gives  w — 4 l\/(,s2  4"  10s  + 9)  — a + 3], 

4 


must  be  a max. 

TV3  - S3V)3 

iu  fluxions  and  reduced. 


Here  if  s = 1,  ay 


1 + V* 


but  if  s be  diminished  without 


limit,  w = | s ; if  it  be  augmented  without  limit,  then  will 
v/(s3  + !0s  + 9)  approach  indefinitely  near  to  s 4*  5,  and 
consequently  w = 2s.  Whence  the  truth  of  the  proposition 
is  manifest. 

PROPOSITION  IX. 

Let  tp  denote  the  absolute  effort  of  any  moving  force,  ■when 
it  has  no  velocity  ; and  suppose  it  not  capable  of  any  effort 
when  the  velocity  is  w ; let  f be  the  effort  answering  to  the 
velocity  v ; then , if  the  force  be  uniform,  _ f will  be  = 

*(1  - ~)2- 

v w 

For  it  is  the  difference  between  the  velocities  w and  v 
which  is  efficient,  and  the  action,  being  constant,  will  vary  as 
the  square  of  the  efficient  velocity.  Hence  we  shall  have  this 
analogy,  : f : : (w  — 0)2  :|(w  — v)2  : consequently,  f = 


*( 


w ' w 

Though  the  pressure  of  an  animal  is  not  actually  uniform 
during  the  whole  time  of  its  action,  yet  it  is  nearly  so  : so 
that  in  general  we  may  adopt  this  hypothesis  in  order  to  ap- 
proximate to  the  true  nature  of  animal  action.  On  which 
supposition  the  preceding  prop,  as  well  as  the  remaining  one, 
in  this  chapter  will  apply  to  animal  exertion. 

Cor  Retaining  the  same  notation,  we  have  w = — — . 

. ! . . >/<*- ✓ s' 

This,  applied  to  the  motion  of  animals,  gives  this  theorem  : 
The  utmost  velocity  with  which  an  animal  not  impeded  can 
move,  is  to  the  velocity  with  which  it  moves  when  impeded  by 
a given  resistance,  as  the  square  root  of  its  absolute  force,  to 
. the  difference  of  the  square  roots  of  its  absolute  and  ffcient 
forces. 

PROPOSITION  X. 

To  investigate  expressions  by  means  of  which  the  maxi- 
mum effect,  in  machines  whose  motion  is  uniform,  may  be  de- 
termined. 

I.  It  follows  from  the  observations  made  in  art.  1 and  the 
definitions  in  this  chapter,  that  when  a machine,  whether 
; simple  or  compound,  is  put  into  motion,  the  velocities  of  the 

impelled 


428 


MAXIMUM  IN  MACHINES. 


impelled  aDd  working  points,  are  inversely  as  the  forces  which 
are  in  equilibrio,  when  applied  to  those  points  in  the  direc 
tion  of  their  motion.  Consequently,  iff  denote  the  resistance 
wLen  reduced  to  the  working  point,  and  v its  velocity  ; w hile 
f and  v denote  the  force  acting  at  the  impelled  point,  and  its 
velocity  ; we  shall  have  fv  = fo , or  introducing  t the  lime, 
f vt  = fvt.  Hence,  in  all  working  machines  which  haze  ac- 
quired a uniform,  motion , the  performance  of  the  machine  is  equal 
to  the  moment um  of  impulse. 

11.  Let.  f be  the  effort  of  a force  on  the  impelled  point  of 
a machine  when  it  moves  with  the  velocity  v,  the  velocity 
being  w when  f = 0,  and  let  the  relative  velocity  w — v = u. 

Then  since  (prop,  ix)  f = < p (— — -)2,  the  momentum  of  im- 


pulse fv  will  become  v<p  (-)2  = <p  . — - (w  — «)  ; because 

v = w — u.  Making  this  expression  for  fv  a maximum, 
or,  suppressing  the  constant  quantities,  and  making  u2  fv  — u) 
a max.  or  its  dux  — 0,  when  u is  variable,  we  find  2w  = 3m, 
or  u — fw-  Whence  v = w — u — w — f w = iw. 

Consequently,  when  the  ratio  of  v to  v-  is  given,  by  the 
construction  of  the  machine,  and  the  resistance  is  susceptible 
of  variation,  we  must  load  the  machine  more  or  less  till  the 
velocity  of  the  impelled  point,  is  one  third  of  the  greatest  ve- 
locity i f the  force  ; then  will  the  work  done  be  a maximum. 

Or.  the  work  dune  by  an  animal  is  greatest,  when  the  velocity 
with  which  it  moves,  is  one-third  of  the  greatest  velocity  with 
which  it  is  capable  of  moving  when  not  impeded. 

7(2  A\y  2 J 

III.  Since  f — <? — =a(- — ) = a <b,  in  the  case  of  the 

w2  ' w- 

maximum  we  have  fv  = ±(pv  = w = #=Aw,  for  the 
momentum  of  impulse,  or  for  the  work  done,  when  the  ma- 
chine ii  in  its  best  state.  Consequently,  when  the  resistance 
is  a given  quantity,  we  must  make  v : v : : 9f  : 4tp  ; and  this 
Structure  of  the  machine  will  give  the  maximum  effect  — 

IV.  If  we  enquire  the  greatest  effect  on  the  supposition 
that  <p  only  is  variable,  we  must  make  it  infinite  in  the  above 
expression  for  the  work  done,  which  would  then  become 

v v 

wf,  or  vv  -f  or  w - ft,  including  the  lime  in  the  formula. 


Hence  we  see,  that  the  sum  of  the  agents  employed  to  move  a 
machine  >,  ay  be  infinite,  while  the  effect  is  fnite  : j’or  the  varia- 
tions of  <p.  which  are  proportional  to  this  sum,  do  not  influence, 
the  above  expression  lor  ttie  effect. 


Scholium 


MAXIMUM  IN  MACHINES. 


4'29 


Scholium. 

The  propositions  now  delivered  contain  the  most  material 
principles  in  the  theory  of  machines.  The  manner  of  apply- 
ing several  of  them  is  very  obvious  : the  application  of  some, 
being  less  manifest,  may  be  briefly  illustrated,  and  the  chapter 
concluded  with  two  or  three  observations. 

The  last  theorem  may  be  applied  to  the  action  of  men  and 
of  horses,  with  more  accuracy  than  might  at  first  be  sup- 
posed. Observations  have  been  made  on  men  and  horses 
drawing  a lighter  aloDg  a canal,  and  working  several  days 
together.  The  force  exerted  was  measured  by  the  curva- 
ture and  weight  of  the  track  rope,  and  afterwards  by  a spring 
steelyard.  The  product  of  the  force  thus  ascertained,  into 
the  velocity  per  hour,  was  considered  as  the  momentum.  In 
this  way  the  action  of  men  was  found  to  be  very  nearly  as 
(w— v)2  : the  action  of  horses  loaded  so  as  not  to  be  able  to 

_9 

trot  was  nearly  as  (w  — v)  1 1 , or  as  (w  — v)5.  Hence  the 
hypothesis  we  have  adopted  may  in  many  cases  be  safely  as- 
sumed. 

According  to  the  best  observations,  the  force  of  a man  at 
rest  is  on  the  average  about  70  pounds  ; and  the  utmost  velo- 
city with  which  he  can  walk  is  about  6 feet  per  second,  takers 
at  a medium.  Hence,  in  our  theorems,  <p  — 70,  and  w==6. 
Consequently  f = f 0 = 31  £ lbs.  the  greatest  force  a man 
can  exert  when  in  motion  : and  he  wall  then  move  at  the  rate 
of  }w,  or  2 feet  per  second,  or  rather  less  than  a mile  and  a 
half  per  hour. 

The  strength  of  a horse  is  generally  reckoned  about  6 times 
that  of  a man  ; that  is,  nearly  4201bs.  at  a dead  pull.  His  ut- 
most walking  velocity  is  about  10  feet  per  second.  Therefore 
his  maximum  action  will  be  £ of  420  = 186|  lbs.  and  he  will 
then  move  at  the  rate  of  £ of  10,  or  3k  feet,  per  second,  or 
nearly  2£  miles  per  hour.  In  both  these  instances  we  sup- 
pose the  force  to  be  exerted  in  drawing  a weight  along  a ho- 
rizontal plane  ; or  by  raising  a weight  by  a cord  running 
over  a pulley,  which  makes  its  direction  horizontal. 

2.  The  theorems  just  given  may  serve  to  show,  in  what 
points  of  view  machines  ought  to  be  considered,  by  those  who 
would  labour  beneficially  for  their  improvement. 

The  first  object  of  the  utility  of  machines  consists  in  fur- 
nishing the  means  of  giving  to  the  moving  force  the  most  com- 
modious direction  ; and,  when  it  can  be  done,  of  causing  its 
action  to  be  applied  immediately  to  the  body  to  be  moved. 
These  can  rarely  be  united  : but  the  former  can  be  accom- 
plished in  most  instances  ; of  which  the  use  of  the  simple 

lever 


MAXIMUM  IN  MACHINES. 


4 Jo 

lever,  pulley,  and  wheel  and  axle,  furnish  many  examples. 
The  second  object  gained  by  the  use  of  machines,  1-  un  nc-  ; 
c ommodation  of  the  velocity  of  the  work  to  be  performed.,  to  the  i 
velocity  with  which  alone  a natural  power  con  act  Thus  when- 
ever the  natural  power  acts  with  a certain  velocity  which  can-  i 
not  be  changed,  and  the  work  must  be  performed  with  a great-  ! 
er  velocity,  a machine  is  interposed  moveable  round  a fixed 
support,  and  the  distances  of  the  impelled  and  working  points 
are  taken  in  the  proportion  of  the  two  given  velocities. 

But  the  essential  advantage  of  machines,  that,  in  fact,  which 
properly  appertains  to  the  theory  of  mechanics,  consists  in 
augmenting,  or  rather  in  modifyiug,  the  enegy  of  the  mov-  i 
ing  power,  in  such  manner  that  it  may  produce  effects  ol  which 
it  would  have  been  otherwise  incapable.  Thus  a man  might  1 
carry  up  a flight  ot  steps  20  pieces  of  stone,  each  weighing  ■ 
30  pounds  (one  by  one)  in  as  small  a time  as  he  could  (with  i 
the  same  labour)  raise  them  all  together  by  a piece  of  ma-  I 
chinery,  that  would  have  the  velocities  of  the  impelled  and  > 
working  points  as  20  to  1 ; and  in  this  case,  the  instiument 
would  furnish  no  real  advantage,  except  that  of  saving  bis 
steps.  But  if  a large  block  of  20  times  30,  or  GOt'lbs.  w.  ight 
were  to  be  raised  to  the  same  height,  it  would  far  surpass  the  1 
utmost  efforts  of  the  man,  w’ithout  the  intervention  of  some 
such  contrivance, 

The  same  purpose  may  be  illustrated  somewhat  differently  ; 
confining  the  attention  all  along  to  machines  whose  motion 
is  uniform.  The  product  fv  represents,  during  the  unit  of 
time,  the  effect  which  results  from  the  motion  of  the  resist- 
ance ; this  motion  being  produced  in  any  manner  whatever. 

If  it  be  produced  by  applying  the  moving  force  immediately 
to  the  resistance,  it  is  necessary  not  only  that  the  products  fv 
and  fv  should  be  equal  ; but  that  at  the  same  time  f = /,  and 
v = n : if,  therefore,  as  most  frequently  happens,/ be  great- 
er than  f,  it  will  be  absolutely  impossible  to  put  the  resistance 
in  motion  by  applying  the  moving  fprce  immediately  to  it. 
Now  machines  furnish  the  means  of  disposing  the  product  fv 
in  such  a manner  that  it  may  always  be  equal  to  fv,  however 
much  the  factors  of  fv  may  differ  from  the  analogous  factors 
in fvj  and,  consequently,  of  putting  the  system  in  motion, 
whatever  is  the  excess  of / over  f. 

Or,  generally,  as  M.  Prony  remarks  (Archi.  Hydraul.  art. 
504),  machines  enable  us  to  dispose  the  factors  of  fv!  in  such 
a manner,  that  while  that  product  continues  the  same,  its  fac- 
tors may  have  to  each  other  any  ratio  we  desire.  If,  for  in- 
stance, time  be  precious,  the  efi'ect  must  be  produced  in  a very 

short 


MAXIMUM  IN  MACHINES. 


431 


short  time  and  yet  we  should  have  at  command  a.  force  capa- 
ble of  iittie  velocity  but  of  great  effort,  a-  machine  must  be 
found  to  supply  the  velocity  necessary  for  toe  ialens'ty  of  the 
force  ; if,  on  the  contrary , the  mechanist  has  only  a weak  pow- 
er at  his  disposition,  but  ;apat  le  of  a great  velocity,  a machine 
must  be  adopted  that  will  compensate,  by  the  velocity  the 
agent  can  communicate  to  it,  for  the  force  wanted  :<Jastiy.  if 
the  agent  is  capable  neither  of  great  effort,  nor  of  great  velo- 
city a convenient  machine  may  still  enable  him  to  accomplish 
the  effect  desired,  and  make  the  product  FVt  of  force,  veloci- 
ty and  time,  as  great  as  is  requisite.  Thus,  to  give  another 
; example  : Suppose  that  a man  exerting  his  strength  imme- 
diately on  a mass  of  25  lbs,  can  raise  it  vertically  with  a velo- 
city of  4 feet  per  second  ; the  same  man  acting  on  a mass  of 
1000  lbs.  cannot  give  it  any  vertical  motioD  though  he  exerts 
his  utmost  strength  unless  he  has  recourse  to  some  machine. 
Now  he  is  capable  of  producing  an  effect  equal  to  25  X 4 X 
t : the  letter  t being  introduced  because,  if  the  labour  is  con- 
tinued the  value  of  t will  not  be  indefinite,  but  comprised 
within  assignable  limits.  Thus  we  have  25  X 4 X t = 1000 
X v X t ; and  consequently  v ~ TV  of  a foot.  This  man  may 
therefore  with  a machine,  as  a lever,  or  axis  in  peritrochio, 

; cause  a mass  of  lOOOlbs  to  raise  T'T  of  a foot,  in  the  same  time 
that  he  could  raise  25  lbs.  4 feet  without  a machine  ; or  he 
may  raise  the  greater  height  as  far  as  the  less,  by  employing 
40  times  as  much  time. 

From  what  has  been  said  on  the  extent  of  the  effects  which 
may  be  attained  by  machines,  it  will  be  seen  that,  so  long  as 
a moving  force  exercises  a determinate  effort,  with  a velocity 
also  determinate,  or  so  long  as  the  product  of  these  is  con- 
stant, the  effect  of  the  machine  will  remain  the  same  : thus, 
under  this  point  of  view,  supposing  the  preponderance  of  the 
effort  of  the  moving  power,  and  abstracting  from  inertia  and 
friction  of  materials,  the  convenience  of  application,  &c  all 
machines  are  equally  perfect.  But  from  what  has  been 
shown,  (props.  9,  10)  a moving  force  may,  by  diminishing 
its  velocity,  augment  its  effort,  and  reciprocally.  There  is 
therefore  a certain  effort,  of  the  moving  force,  such  that  its 
product  by  the  velocity  which  comports  to  that  effort,  is  the 
greatest  possible.  Admitting  the  truth  of  the  law  assumed 
in  the  propositions  just  referred  to,  we  have,  when  the  effect, 
is  a maximum,  v = |w,  or  f = a <p  ; and  these  two  values 
obtaining  together,  their  product  0w  expresses  the  value 
of  the  greatest  effect  with  respect  to  the  unit  of  time.  In 
practice  it  will  always  be  adviseahle  to  approach  as  nearly  to 
these  values  as  circumstances  will  admit:  fur  it  cannot  be 

expected 


432 


PRESSURE  OF  EARTH  AND  FLUIDS. 


expected  that  they  can  always  he  exactly  attained.  But  a 
small  variation  will  not  be  of  much  consequence  : for,  by  a 
well-known  property  of  those  quantities  which  admit  of  a 
proper  maximum  and  minimum,  a value  assumed  at  a mode- 
rate distance  from  either  of  these  extremes  will  produce  no 
sensible  change  in  the  effect. 

If  th#  relation  of  f to  v followed  any  other  law  than  that 
which  we  have  assumed,  we  should  6nd  from  the  expression 
of  that  law  values  of  f,  v,  &c.  different  from  the  preceding. 
The  general  method  however  would  be  nearly  the  same. 

With  respect  to  practice,  the  grand  object  in  all  cases 
should  be  to  procure  a uniform  motion , because  it  is  that  from 
which  (cceteWs  paribus')  the  greatest  effect  always  results. 
Every  irregularity  in  the  motion  wastes  some  of  the  impelling 
power  : and  it  is  the  greatest  only  of  the  varying  velocities 
which  is  equal  to  that  which  the  machine  would  acquire  if  it 
moved  uniformly  throughout  : for,  while  the  motion  accele- 
rates, the  impelling  force  is  greater  than  what  balances  the  re- 
sistance at  that  time  opposed  to  it,  and  the  velocity  is  less  than 
what  the  machine  would  acquire  if  moving  uniformly  ; and 
when  the  machine  attains  its  greatest  velocity,  it  attains  it  be- 
cause the  power  is  not  then  acting  against  the  whole  resistance. 
In  both  these  situations  therefore,  the  performance  of  the  ma- 
chine is  less  than  if  the  power  and  resistance  were  exactly 
balanced;  in  which  case  it  would  move  uniformly  (art.  1.) 
Besides  this,  when  the  motion  of  a machine,  and  particularly 
a very  ponderous  one,  is  irregular,  there  are  continual  repeti- 
tions of  strains  and  jolts  which  soon  derange  and  ultimately 
destroy  the  whole  structure.  Everj  attention  should  there- 
fore be  paid  to  the  removal  of  all  causes  of  irregularity. 


PRESSURE  OF  EARTH  AND  FLUIDS  AGAINST  WALLS  ANP 
FORTIFICATIONS,  THEORY  OF  MAGAZINES,  &c. 

PROBLEM  I. 

To  determine  the  Pressure  of  Earth  against  Walls. 

When  new-made  earth,  such  as  is  used  in  forming  ram- 
parts, &c.  is  not  supported  by  a wall  as  a facing,  or  by  coun- 
terforts and  land-ties,  &c  but  left  to  the  action  of  its  weight 
and  the  weather  ; the  particles  loosen  and  separate  from  each 

other. 


AGAINST  WALLS,  kc. 


43 2 


either,  and  form  a sloping  surface,  nearly  regular  ; which 
plane  surface  is  called  the  natural  slope  of  the  earth  ; and  is 
supposed  to  have  always  the  same  inclination  or  deviation 
| from  the  perpendicular,  in  the  same  kind  of  soil.  In  com- 
mon earth  or  mould,  being  a mixture  of  all  sorts  thrown  to- 
gether, the  natural  slope  is  commonly  at  about  half  a right 
angle,  or  45  degrees  ; but  clay  and  stiff  loam  stands  at  a greater 
angle  above  the  horizon,  while  sand  and  light  mould  will  only 
stand  at  a much  less  angle.  The  engineer  or  builder  must 
therefore  adapt  his  calculations  accordingly. 

Now.  we  have  already  given,  (at  prop.  45  Statics)  the 
general  theory  and  determination  of  the  force  with  which 
the  triangle  of  earth  (which  would  slip  down  if  not  sup- 
ported) presses  against  the  wall  on 
the  most  unexceptional  principles, 
acting  perpendicularly  against  ae  at  k, 
or  i of  the  altitude  ae  above  the  foun- 
dation at  e ; the  expression  for  which 

iforce  was  there  found  to  be~--E-  ^ 

[ 6Ee2  ’ 

where  m denotes  the  specific  gravity  of 
the  earth  of  the  triangle  abe. — It  may  be  remarked  that  this 
was  deduced  from  using  the  area  only  of  the  profile,  cr  trans- 
verse triangular  section  abe,  instead  of  the  prismatic  solid  of 
my  given  length,  haying  that  triangle  for  its  base.  Arfti  the 
same  thing  is  done  in  determining  the  power  of  the  wall  to 
support  the  earth,  viz.  using  only  its  profile  or  transverse  sec- 
tion in  the  same  plane  or  direction  as  the  triangle  abe.  This 
:t  is  evident  will  produce  the  same  result  as  the  solids  them- 
selves, since,  beiDg  both  of  the  same  given  length,  these  have 
the  same  ratio  as  their  transverse  sections. 

In  addition  to  this  determination,  we  may  here  further  ob- 
serve, that  this  pressure  ought  to  diminished  in  proportion 

0 the  cohesion  of  the  matter  in  sliding  down  the  inclined 
ilane  be.  Now  it  has  been  found  by  experiments,  that  a 
jody  requires  about  one-third  of  its  weight  to  move  it  along 

1 plane  surface.  The  above  expression  must  therefore  be 
reduced  in  the  ratio  of  3 to  2 ; by  which  means  it  becomes 

■ - ' — m for  the  .true  practical  efficacious  pressure  of  the 

5arth  against  the  wall. 

Since  — , which  occurs  in  this  expression  of  the  force  of 
Be 

he  earth,  is  equal  to  the  sine  of  the  ^ aeb  to  the  radius  1, 
iut  the  sine  of  that  /_  e = e ; also  put  a — ae  the  altitude 
>f  the  triangle  : then  the  above  expression  of  the  force,  viz. 

Vol.  II.  5fi  , ae3  . AB® 


434 


PRESSURE  OF  EARTH  AND  FLUIDS 


AE3  ’ AB2 


2 -m, becomes  £a3e2m,  for  the  perpendicular  pressure  of 

the  earth  against  the  wall.  And  if  that  aDgle  be  45°,  a?  is 
usually  the  case  in  common  earth,  then  is  e 2 = and  the 


pressure  becomes  T\a3m. 


PROBLEM  II. 


To  determine  the  Thickness  of  Wall  to  support  the  Earth. 


In  the  first  place  suppose  the  section 
of  the  wall  to  be  a rectangle,  or  equally 
thick  at  top  and  bottom,  and  of  the  same 
height  as  the  rampart  of  earth,  like  aefg 
in  the  annexed  figure.  Conceive  the 
weight  w,  proportional  to  the  area  ge, 
to  be  appended  to  the  base  directly  be- 
low the  centre  of  gravity  of  the  figure. 


E jRT 
vrw 


Now  the  pressure  of 
the  earth  determined  in  the  first  problem,  being  in  a direction 
parallel  to  ag,  to  cause  the  wall  to  overset  and  turn  back 
about  the  point  f,  the  effort  of  the  wall  to  oppose  that  effect, 
will  be  the  weight  w drawn  into  fn  the  length  of  the  lever 
by  which  it  acts,  that  is  w X fn,  or  aefg  X fn  in  general, 
whatever  be  the  figure  of  the  wall. 

But  now  in  case  of  the  rectangular  figure,  the  area  ge  = ae 
X ef  = ax,  putting  a = ae  the  altitude  as  before,  and  x — ef 
the  required  thickness  ; also  in  this  case  fn  = Lef  = ±x,  the 
centre  of  gravity  being  in  the  middle  of  the  rectangle.  Hence 
then  ax  X \x  = \ax2 , or  rather  ±ax-n  is  the  effort  of  the 
wall  to  prevent  its  being  overturned,  n denoting  the  specific 
gravity  of  the  wall. 

Now  to  make  this  effort  a due  balance  to  the  pressure  of 
the  earth,  we  put  the  two  opposing  forces  equal  that,  is 
\ax2n—±a3e2m,  or  ±x2n  = ±a2e2m,  an  equation  which  gives 
2wz>  . • 

x—\ ae^/  — , for  the  requisite  thickness  of  the  wall,  just 


to  sustain  it  in  equilibrio. 


Corol.  1.  The  factor  ae,  in  this  expression,  is  = the  line 
aq  drawn  perp.  to  the  slope  of  earth  be  : theref.  the  breadth 

x becomes  = i Aft  — , which  conseq.  is  direct^  propor- 
tional to  the  perp.  Aft. — When  the  angle  at  e is  = 45°,  or 
half  a right  angle,  as  is  commonly  the  case,  its  sine  e is^^/ 

and  the  breadth  of  the  wall  x = Ac*/  — . Further,  when  the 


wall  is  of  brick,  its  specific  gravity  is  nearly  the  same  as 

the 


AGAINST  WALLS,  &c. 


435 


the  earth,  or  m = n,  and  then  its  thickness  x = ia,  or  one- 
third  of  its  height. — But  when  the  wall  is  of  stone,  of  the 
specific  gravity  21,  that  of  earth  being  nearly  2,  that  is, 

m — - 2,  and  n — 21  ; then  y/  y/  i — *895,  g-  of  which 

is  *298,  and  the  breadth  x ~ -298a  = r\a  nearly.  That  is, 
the  thickness  of  the  stone  wall  must  be  y3^  of  its  height. 

PROBLEM  1IL 

To  determine  the  Thickness  of  the  Wall  at  the  Bottom , when, 
its  Section  is  a Triangle , or  coming  to  an  Edge  at  Top. 

In  this  case,  the  area  of  the  wall  aef  CB A. 

is  only  half  of  what  it  was  before,  or 
only  i ae  X ef  = lax,  and  the  weight 
w = \axn.  But  now,  the  centre  of 
gravity  is  at  only  i of  fe  from  the  line 
ae  , or  fn  = | fe  = \x.  Consequently 
fk  X w = |x  X i axn  = \ax 2n.  This, 
as  before,  being  put  = the  pressure  of 
the  earth,  gives  the  equ ation  i ax  2n  = ±a3e2moT,x2n  — la2  e2  m, 

and  the  root  x,  or  thickness  EF  = aev/  — = — for  the 

3n  6 n 

slope  of  45°. 

bJow  when  the  wall  is  of  brick,  or  m — n nearly,  this  be- 
comes = *408a  = | o,  or  tl  of  the  height  nearly. 

But  when  the  wall  is  of  stone,  or  m to  n as  2 to  2i,  then 

y/™  y/~  f , and  the  thickness  x or  a y/-~-  — a */  T2 j = ‘365a 
= |a  nearly,  or  nearly  f of  the  height. 


PROBLEM  IV. 


To  determine  the  Thickness  of  the  Wall  at  the  Top,  when  the 
Face  is  not  Perpendicular,  but  Inclined  as  the  Front  of  a Forti- 
fication Wall  usually  is. 

Here  gf  represents  the  outer  face  of 
a fort,  aefg  the  profile  of  the  wall, 
having  ag  the  thickness  at  top,  and  ef 
that  at  the  bottom.  Draw  gh  prep,  to 
ef  5 and  conceive  the  two  weights  w, 
w,  to  be  suspended  from  the  centres  of 

I gravity  of  the  rectangle  ah  and  the  tri- 
angle ghf,  and  to  be  proportional  to 
their  areas  respectively.  Then  the  two  momenta  of  the 
weights  w,  w,  acting  by  the  levers  fn,  fm,  must  be  made  equal 


| to  the  pressure  of  the  earth  in  the  direction  prep,  to  ae. 


Now 


436 


PRESSURE  OF  EARTH  AND  FLUIDS 


Now  put  the  required  thickness  ag  or  eh  = x,  and  the 
altitude  ae  or  gh  = a as  before.  And  because  in  such  cases 
the  slope  of  the  wall  is  usually  wade  equal  to  a of  its  altiiude; 
that  is  fh  — }isor  ja,  the  lever  fm  will  be  § of  ia  = f^a, 
and  the  lever  fn  = fh  -f  Aeh  = }a  + \x.  But  the  area 
of  ghf  — gh  X 4hf  — a X j1  7 o = T‘jfi2  = w,  and  the  area 
ah  — ae  x ag  = ax  — xv  ; these  two  drawn  into  the  re- 
spective levers  fm,  fn,  give  the  two  momenta,  J-sauu=-f-sa.  x 
TVa2  = Vj0®’  an^  (ia  4"  4*0  X ax  — a a-x  + a a* 2 ; theref. 
the  sum  of  the  two,  (|ax2  -f-  4a2  x + Jy  a3)n  must  be  =Aya3/n, 

or  dividing  by  ton,  x2  -j-  §ax  + y2y  a2  — la2  x — ; now  adding 
i fja2  to  both  sides  to  complete  the  square,  the  equation 
becomes. r2  -f- § ax -f  yV ft2  = £a2  + ^a2 , the  root  of  which 

is  x + {-a— a y/  (5V  + ^4>nd  hence  x = a y/^'j  + ia 

And  the  base  ef  — a y/  ( Jy  + 

Now,  for  a brick  wall,  m—n  nearly,  and  then  the  breadth 
x — a y/  (yL  -f  i)  — jti  = i-gdy/  34  — \a  — 1 89a,  or  almost 

Aa  in  brick  walls. — But  the  stone  walls,  — = i,  and  x = 

3 n s 

0 </  (is  + t4j)  — }a  = A_a  y/  29, — Aa  ==159#=  fjO  nearly... 
for  the  thickness  ag  at  the  top,  in  stone  walls. 

in  the  same  manner  we  may  proceed  when  the  slope  is 
supposed  to  be  any  other  part  of  the  altitude,  instead  of  a as 
used  above.  Or  a general  solution  might  be  given,  by  as- 
suming the  thickness  = ^ part  of  the  altitude. 

REMARK. 

Thus  then  we  have  given  all  the  calculations  that  may  be 
necessary  in  determining  the  thickness  of  a wall,  proper  to 
support  the  rampart  or  body  of  earth,  in  any  work  If  it 
should  be  objected,  that  our  determination  gives  only  such  a 
thickness  of  wall,  as  makes  it  an  exact  mechanical  balance  to 
the  pressure  or  push  of  the  earth,  instead  of  giving  the 
former  a decided  preponderance  over  the  latter,  as  a security 
against  any  failure  or  accidents.  To  this  we  answer,  tha; 
what  has  been  done  is  sufficient  to  insure  stability,  for  the 
following  reasons  and  circumstances.  First,  it  is  usual  to 
build  several  counterforts  of  masonry,  behind  and  against  the 
wall,  at  certain  distances  or  intervals  from  one  another  : which 
contribute  very  much  to, strengthen  the  wall,  and  to  resist  the 
pressure  of  the  rampart  2dly  We  have  omitted  to  include 
the  effect  of  the  parapet  raised  above  the  wall  ; which  must 
add  somewhat,  by  its  weight,  to  the  force  or  resistance  of  the 

walk 


AGAINST  WALLS,  &c. 


43? 


wall.  It  is  true  we  could  have  brought  these  two  auxiliaries 
to  exact  calculation,  as  easily  as  we  have  done  for  the  wa]l 
itself  : but  we  have  thought  it  as  w<dl  to  leave  these  two  ap- 
pendages, thrown  in  as  indeterminate  additions,  above  the 
exact  balance  of  the  wall  as  before  determined,  to  give  it  an 
assured  stability.  Besides  these  advantages  in  the  wall  itself, 
certain  contrivances  are  also  usually  employed  to  diminish 
the  pressure  of  the  earth  against  it  : such  as  land-ties  and 
branches,  laid  in  the  earth,  to  diminish  its  force  and  push 
against  the  wall  For  all  these  reasons  then,  we  think  the 
practice  of  making  the  wall  of  the  thickness  as  assigned  by 
our  theory,  may  be  safely  depended  on,  and  profitably  adopted  ; 
as  the  additional  circumstances,  just  mentioned,  will  suffi 
ciently  insure  stability  ; and  its  expense  will  be  less  than  is  in 
eurred  by  any  former  theory. 

PROBLEM  Y. 

j , To  determine  the  Quantity  of  Pressure  sustained  by  a Dam  or 
Sluice,  made  to  pen  up  a Body  of  Water. 

By  art.  313  Hydrostatics,  (in  this  volume)  the  pressure  of 
a fluid  against  any  upright  surface,  as  the  gate  of  a sluice  or 
canal,  is  equal  to  half  the  weight  of  a column  of  ihe  fluid, 
whose  base  is  equal  to  the  surface  pressed,  and  its  altitude 
the  same  as  that  of  the  surface.  Or,  by  art  314  of  the  same, 
the  pressure  is  equal  to  the  weight  of  a column  of  the  fluid, 
whose  base  is  equal  to  the  surface  pressed,  and  its  altitude 
equal  to  the  depth  of  the  centre  of  gravity  below  the  top  or 
i surface  of  the  water  ; which  comes  to  the  same  thing  as  the 
; former  article,  when  the  surface  pressed  is  a rectangle,  be- 
i cause  its  centre  of  gravity  is  at  half  the  depth.  . 

Ex.  1.  Suppose  the  dam  or  sluice  be  a rectangle  whose 
length,  or  breadth  of  the  canal,  is  20  feet,  and  the  depth  of 
j vyater  6 feet.  Here  20  X 6 = 120  feet,  is  the  area  of  the 
surface  pressed  ; and  the  depth  of  the  centre  of  gravity  being 
3 feet,  viz.  at  the  middle  of  the  rectangle  ; therefore  120  X 
3 = 360  cubic  feet  is  the  content  of  the  column  of  water. 
But  each  cubic  foot  of  water  weighs  1000  ounces,  or  62i 
pounds  ; therefore  360  X 1000  = 360000  ounces,  2250O 
pounds,  or  10  tons  and  100  lb,  is  the  weight  of  the  column  of 
water,  or  the  quautity  of  pressure  on  the  gate  or  dam. 

Ex.  2.  Suppose  the  breadth  of  a canal  at  the  top,  or  sur- 
face of  the  water,  to  be  24  feet,  but  at  the  bottom  only  16 
feet,  the  depth  of  water  being  6 feet,  as  in  the  last  example  : 
required  the  pressure  on  a gate  which,  standing' across  the 
fanal,  dams  the  water  up  ? 


Here 


438 


PRESSURE  OF  EARTH  AND  FLUIDS 


Here  the  gate  is  in  form  of  a trapezoid, 
having  the  two  parallel  sides  ab,  cd,  viz. 
ab  = 24,  and  cd  = 16,  and  depth  6 feet. 

Now,  by  mensuration  problem  3,  volume  1, 
i(ab  +cd)  X 6,  = 20  X 6 = 120  the  area 
of  the  sluice,  the  same  as  before  in  the  1st 
example  : but  the  centre  of  gravity  caonot 
be  so  low  down  as  before,  because  the  figure 
is  wider  above  and  narrower  below,  the  whole 
depth  being  the  same. 

Now,  to  determine  the  centre  of  gravity 
k of  the  trapezoids  ad,  produce  the  two 
sides  ac,  bd,  till  they  meet  in  c ; also  draw  gkf.  and  cn 

perp.  to  ab  : then  ah  : ch  : : ae  : ge,  that  is,  4 : 6 : : 12  : 

18  = ge  ; and  ef  being  = 6,  theref.  fg  = 12.  Now,  by 
Statics  art.  229,  ef  = 6 = Aeg  gives  f the  centre  of 
gravity  of  the  triangle  abg,  and  fi  = 4 = afg  gives  i the 
centre  of  gravity  of  the  triangle  cdg.  Then  assuming  k to 
denote  the  centre  of  ad,  it  will  be,  by  art.  212  this  vol.  as  the 
trap,  ad  : A cdg  : : if  : fk,  or  A abc—  A cdg  : A cdg  : : 

if  : fk,  or  by  theor.  88  Geom.  ge2  — gf2  : gf2  : : if  : fk, 

that  is  182  — 122  to  122  or  32  — 22  to  22  or  5 : 4 : : if  = 4 : 
y = 3i  = pk  ; and  hence  ek  = 6 — 31  = 2|  = y is  the 
distance  of  the  centre  k below  the  surface  of  the  water  This 
drawn  into  120  the  area  of  the  dam-gate,  gives  336  cubic  feet 
of  water  = the  pressure,  rr:  336000  ounces  = 21000  pounds 
= 9 tons  80  lb,  the  quantity  of  pressure  against  the  gate,  as 
required,  being  a 15th  part  less  than  in  the  first  case. 

Ex.  3.  Find  the  quantity  of  pressure  against  a dam  or 
sluice,  across  a canal,  which  is  20  feet  wide  at  top,  14  at  bot- 
tom, and  8 feet  depth  of  water  ? 

PROBLEM  VI. 

To  determine  the  Strongest  Angle  of  Position  of  a Pair  of  Gates 
for  the  Lock  on  a Canal  or  River. 

Let  ac,  bc  he  the  two  gates,  meet- 
ing in  the  angle  c,  projecting  out 
against  the  pressure  of  the  water,  ab 
being  the  breadth  of  the  canal  or  river. 

Now  the  pressure  of  the  water  on  a 
gate  ac,  is  as  the  quantity,  or  as  the 
extent  or  length  of  it,  ac.  And  the  mechanical  effect  of  that 
pressure,  is  as  the  length  of  lever  to  the  middle  of  ac,  or  as 
ac  itself.  On  both  these  accounts  then  the  pressure  is  as 

AC3, 


AGAINST  WALLS,  kc. 


439 


ac2.  Therefore  the  resistance  or  the  strength  of  the  gate 
must  be  as  itie  reciprocal  of  this  ac3. 

Now  produce  ac  to  meet  bd,  prep,  to  it,  in  d ; and  draw 
ce  to  bisect  ae  perpendicularly  in  e ; then,  by  similar  tri- 
angles, as  ac  : ae  . : ae  : ad  ; where,  ae  and  ae  being  given 
lengths,  ad  is  reciprocally  as  ac,  or  ad2  reciprocally  as  ac2  ; 
that  is,  ad2  is  as  the  resistance  of  the  gate  ac.  But  the  re- 
sistance of  ac  is  increased  by  the  pressure  of  the  other  gate 
in  the  direction  bc.  Now  the  force  in  bc  is  resolved  into  the 
two  bd,  dc  ; the  latter  of  which,  dc,  being  parallel  to  ac, 
has  no  effect  upon  it  ; but  the  former,  bd,  acts  perpendicular- 
ly on  it.  Therefore  the  whole  effective  strength  or  resistance 
of  the  gate  is  as  the  product  ad2  X bd. 

If  now  there  be  put  ab  = a,  and  bd  = x , then  ad2  = ab2 
— bd2  =a2 —x2  ; conseq.  ad2  Xbd  = (a2 — x2')Xx  = a3x  — x3 
for  the  resistance  of  either  gate.  And,  if  we  would  have  this 
to  be  the  greatest,  or  the  resistance  a maximum,  its  fluxion 
must  vanish,  or  be  equal  to  nothing  : thatis,  a2 x — 3x2x  = 0 ; 
hence  az  = 3x2,  and  a:  = a = '57735 a,  the  na- 

tural sine  of  35°  16'  : that  is,  the  strongest  position  for  the 
lock  gates,  is  when  they  make  the  angle  a or  b —35°  16', 
or  the  complemental  angle  ace  or  bce  = 54°  44',  or  the 
whole  salient  angle  acb  = 109°  28‘. 

Scholium. 


Allied  to  this  problem,  are  several  other  cases  in  mechanics, 
such  as,  the  action  of  the  water  on  the  rudder  of  a ship,  in 
sailing,  to  turn  the  ship  about,  to  alter  her  course  ; and  the 
action  of  the  wind  on  a ship’s  sails,  to  impel  her  forward  ; 
also  the  action  of  water  on  the  wheels  of  water  mills,  and  of 
the  air  on  the  sails  of  wind-mills,  to  cause  them  to  turn 
round. 


Thus,  for  instance,  let 
abc  be  the  rudder  of  a 
ship  abde,  sailing  in  the 
direction  bd,  the  rudder 
placed  in  the  oblique  posi- 
tion bc,  and  consequently 
striking  the  water  in  the 
direction  cf,  parallel  to  bd. 


E 

7 


F 

bf  prep. 


to  ec,  and  eg 


C G 
Draw 

prep,  to  cf.  I hen  the  sine  of  the  angle  of  incidence,  of  the- 
direction  of  the  stroke  of  the  rudder  against  the  water,  will 
be  bf,  to  the  radius  cf  ; therefore  the  force  of  the  water 
against  the  rudder  will  be  as  bf2  , by  art.  3,  Mot.  of  bod.  in 
r lui  this  vol.  But  the  force  bf  resolves  into  the  two  eg,  cf,  of 
which  the  latter  is  parallel  to  the  ship’s  motion,  and  therefore 

has 


440 


PRESSURE  OF  EARTH  AND  FLUIDS 


lias  no  effect  to  change  it  ; hut  the  former  bg,  being  prep,  t* 
the  ship’s  motion,  is  the  only  part  of  the  force  to  turn  the  ship 
about  and  change  her  course.  BuIbf  : bg  : : cf  : cb,  there- 
fore cf  : cb  : : bf2  : — — — the  force  upon  the  rudder  te 
- cf  1 

turn  the  ship  about. 

Now  put  a = cf,  x — bc  ; then  bf2  = a2 

BC  . BF2  X (q2  — x2  ) lt~  X — X2 


force 


a;2,  and  the 
and,  to  have  this  a 


maximum,  its  flux,  must  be  made  to  vanish,  that  is,  a2  i — 3x2 
x — 0 ; and  hence  x — a ==  bc  = the  natural  sine  of  35° 

16'  = angle  f ; therefore  the  complemental  angle  c = 
54°  44'  as  before,  for  the  obliquity  of  the  rudder,  when  it  is 
most  efficacious. 


The  ease  will  be  also  the  same  with  respect  to  the  wind 
acting  on  the  sails  of  a wind-mill,  or  of  a ship,  viz.  that  the 
sails  must  be  set  so  as  to  make  an  angle  of  54°  44  with  the 
direction  of  the  wind  ; at  least  at  the  beginning  of  the  mo- 
tion, or  nearly  so  wlien-the  velocity  of  the  sail  is  but  small  in 
comparison  with  that  of  the  wind  ; but  when  the  former  is 
pretty  considerable  in  respect  of  the  latter,  than  the  angle 
ought  to  be  proportionally  greater,  to  have  the  best  effect,  as 
shown  in  Maclaurin’s  Fluxions,  pa  734,  &c- 

A consideration  somewhat  related  to  the  same  also,  is  the 
greatest  effect  produced  on  a mill-wheel,  by  a stream  of  water 
striking  upon  its  sails  or  float-boards.  The  proper  way  in  this 
case  seems  to  be,  to  consider  the  whole  of  the  water  as  acting 
on  the  wheel,  but  striking  it  only  with  the  relative  velocity, 
or  the  velocity  with  which  the  w'ater  overtakes  and  strikes 
upon  the  wheel  in  motion,  or  the  difference  between  the  ve- 
locities of  the  w'heel  and  the  stream.  This  then  is  the  power 
or  force  of  the  water  ; which  multiplied  by  the  velocity  of 
the  wheel,  the  product  of  the  two,  viz  of  the  relative  velo- 
city and  the  absolute  velocity  of  the  wheel,  that  is  (v — u)  v — 
vv  — v2,  will  be  the  effect  of  the  wheel  ; where  v denotes 
the  given  velocity  of  the  water,  and  v the  required  velocity 
of  the  wheel.  Now,  to  make  the  effect  vv  - v2  a maximum, 
or  the  greatest,  its  fluxion  must  vanish,  that  is  \v  — 2v{,  = 0, 
hence  v — -|v  ; or  the  velocity  of  the  wheel  will  be  equal  to 
half  the  velocity  of  the  stream,  when  the  effect  is  the  greatest  ; 
and  this  agrees  best  with  experiments. 

A former  way  of  resolving  this  problem  was,  to  consider 
the  water  as  striking  the  wheel  with  a force  as  the  square  ot 
the  relative  velocity,  and  this  multiplied  by  the  velocity  of 
the  wheel,  to  give  the  effect ; that  is,  (\ — v)2v  — the  effect. 
Now  the  flux,  of  this  product  is  (v — n)2r  — (y-ri>)  X 2v-y  = 0; 

hence 


AGAINST  WALLS,  &c. 


441 


hence  v—v  = 2v,  orv  = 3v , and  v = |v,  or  the  velocity  of 
the  wheel  equal  only  to  k of  the  velocity  of  the  water. 

PROBLEM  VII. 

I To  determine  the  Form  and  Dimensions  of  Gunpowder  Magd- 
' zines. 

In  the  practice  of  engineering,  with  respect  to  the  erection 
of  powder  magazines,  the  exterior  shape  is  usually  made  like 
tlv  roof  of  a house,  having  two  sloping  sides,  forming  two  in- 
clined planes,  to  throw  off  the  rain,  and  meeting  in  an  angle 
or  ridge  at  the  top  ; while  the  interior  represents  a vault, 
more  or  less  extended,  as  the  occasion  may  require  ; and  the 
shape  or  tranverse  section,  in  the  form  of  some  arch,  both 
for  strength  anti  commodious  room,  for  placing  the  powder 
barrels.  It  has  beer,  usual  to  make  this  enterior  curve  a semi- 
| circle.  But,  against  this  shape,  for  such  a purpose,  I must 
enter  ray  decided  protest  ; as  it  is  an  arch  the  farthest  of  any 
t from  being  in  equilibrium  in  itself,  and  the  weakest  of  any, 
by  being  unavoidably  much  thinner  in  one  part  than  in  others. 
Besides  it  is  constantly  found,  that  after  the  centering  of  semi- 
circular arches  is  struck,  and  removed  they  settle  at  the 
crown,  and  rise  up  at  the  flanks,  even  with  a straight  horizon- 
tal form  at  top,  and  still  much  more  so  in  powder  magazines 
with  a sloping  roof  ; which  effects  are  exactly  what  might  be 
expected  from  a contemplation  of  the  true  theory  of  arches. 
Now  this  shrinking  of  the  arches  must  be  attended  with  other 
additional  bad  effects,  by  breaking  the  texture  of  the  cement, 
after  it  has  been  ;n  some  degree  dried  and  also  by  opening  the 
joints  of  the  voussoirs  at  one  end.  Instead  of  the  circular 
arch  therefore,  we  shall  in  this  place  give  an  investigation, 
founded  on  the  true  principles  of  equilibrium,  of  the  only 
just  form  of  the  interior,  winch  is  properly  adopted  to  the 
usual  sloped  roof 

I For  this  purpose,  put  a — dk  the 
hickness  of  the  arch  at  the  top,  x = 
iny  absciss  dp  of  the  required  arch 
idcm.  u — jtr  the  corresponding 
ibseiss  of  the  given  exterior  line  ki, 
ind  y = pc  = ri  their  equal  ordi- 
lates.  Then  by  the  principles  of 
trehes,  in  my  tracts  on  that  subject 
t is  found  that  ci  or  w = a + x — 


i — R X v--- 





<4 


X — , supposing  y a constant 

y*  v 

|uantity,  and  where  q is  some  certain  quantity  to  be  deter- 
uined  hereafter.  But  kr  or  u is  =-  ty,  if  t be  put  to  denote 
Vol.  II.  ' 57  fhe 


•142 


PRESSURE  OF  EARTH  AND  FLUIDS 


the  tangent  of  the  given  angle  of  elevatien  k.ir,  to  radios  1 

and  then  the  equation  is  w =■  a x—ty  — ~ . 

y 2 

Now,  the  fluxion  of  the  equation  IC 

w = a + x — ty,  is  w = x — ty 
and  the  2d  fluxion  is  w — x ; there- 
fore the  foregoing  general  equation  “ 

becomes  w = — ; and  hence  w%u  — 
y- 

~r^—,  the  fluent  of  which  gives  w2  — 

i A Q M 

■p— : but  at  d the  value  of  w is  = a,  and  w = 0,  the  curve 

at  n being  parallel  to  ki  ; therefore  the  correct  fluent  is 

— a3  = 2^1 . Hence  then  y2  — — . 0r  y = — — • 

y2  y W2-a2'  y V'(W2  -a2)  ’ 

the  correct  fluent  of  which  gives  y = ^ x hyp.  log.  of 

*W  + v'Cw2  — as) 
a 

Now,  to  determine  the  value  of  q,  we  are  to  consider  that 
when  the  veitical  line  ci  is  in  the  position  al  or  mn  then 
w — ci  becomes  = al  or  mn  = the  given  quantity  c sup- 
pose, and  y — Aq  or  qm  = b suppose,  in  which  position  the 

last  equation  becomes  b — q X hyp.  log. f t~a  .) ■ an(j 

a 

hence  it  is  found  that  the  value  of  the  constant  quantity 
•v/  is  -fa  [ c _|_  ^ (-co  _„■>)  ; which  being  substituted  for  it,  in 
the  above  general  value  of  y , that  value  becomes 

log. 

u_  ^ ^ log,  of  w + y/  (a>g  — as  )— log.g 

,c+v/(c2  — 02)  log.of  c + v {c2  — a2)  - iog.  a 


y—  b X- 

log.  of - 

from  which  equation  the  value  of  the  ordinate  pc  may  always 
be  found,  to  every  given  value  of  the  vertical  ci. 

But  if,  on  the  other  hand,  pc  be  given,  to  find  ci,  which 
will  be  the  more  convenient  way,  it  may  be  found  in  the 

following  manner  : Put  a <=  log.  of  a,  and  c = ^ X log.  of 

o 

c -f-  Cc2  — a2)  . 

; then  the  above  equation  gives  cy  + a = log. 

of  w -f-  (t t>2  — a2)  ; again,  put  n = the  number  whose  log. 
is  cy  + a ; then  n — w + y/  (a>2  — a2)  ; and  hence  == 

2u~  = CI- 

Now,  for  an  example  in  numbers,  in  a real  case  of  this 

nature. 


AGAINST  WALLS,  &c, 


443 


nature,  let  the  foregoing  figure  represent  a transverse  vertical 
section  of  a magazine  arch  balanced  in  all  its  parts,  in  which 
the  span  or  width  am  is  20  feet,  the  pitch  or  height  dq  is 
10  feet,  thickness  at  the  crown  dk  = 7 feet,  and  the  angle 
of  the  ridge  lks  112°  37',  orthe  half  of  it  lkd  = 56°  18'i, 
the  complement  of  which,  or  the  elevation  kir,  is  33°  41'i, 
the  tangent  of  which  is  = f , which  will  therefore  be  the 
value  of  t in  the  foregoing  investigation.  The  values  of  the 
other  letters  will  be  as  follows,  viz.  dk=«=7  : aq=b  = 10  ; 
dq=A=10  ; al  = c=10i=V  ; a = log.  of  7 = 8450980  ; 


= - Xlog. 


of 


opv'G2 -a3)  _ 


= rV  log-  of 


31-f-^/  520 


a • “ - 21  10 

log.  of  2-56207  = -0408591  ; cy  + a — -0408591  y + 
•8450980  = log.  of  n.  From  the  general  equation  then,  viz. 

az-j-n2  qZ  . ■ . 

a = w — — - = — -f-  hy  assuming  y successively 

equal  to  1,  2,  3,  4,  &c.  thence  finding 
the  corresponding  values  of  cy  -f-  a or 
•0408591y  -f-  -8450980,  and  to  these, 
as  common  Jogs,  taking  out  the  corres- 
ponding natural  numbers,  which  will 
be  the  values  of  n ; then  the  above 
theorem  will  give  the  several  values  of 
w or  ci,  as  they  are  here  arranged  in 
the  annexed  table,  from  which  the 
figure  of  the  curve  is  to  be  constructed, 
by  thus  finding  so  many  points  in  it. 

Otherwise.  Instead  of  making  n 
the  number  of  the  log.  cy  -f-  a,  it  we 
put  in  — the  natural  number  of  the  log. 

, . W+./(>v2 — «2) 

cy  only  ; then  m = , and  am  — .w  = (w2  — 

and 


Val.  of 
or  cp. 

Val.  of  » 
or  ci. 

1 

7-0309 

2 

7-1243 

3 

7-2806 

4 

7-5015 

5 

7-7888  ■ 

6 

8-1452 

7 

8-5737 

8 

9-0781 

9 

9-6623 

10 

10-3333 

a2),  or  by  squaring,  & c.  a2m2  — 2amw-{-wi  = w2  — a 2 

m2  ~f- 1 

hence  w = X a : to  which  the  numbers  being  applied, 

2m 

the  very  same  conclusions  result  as  in  the  foregoing  calculation 
and  table. 


PROBLEM  VIII. 

To  construct  Powder  Magazines  with  a Parabolical  Arch. 

It  has  been  shown,  in  my  tract  on  the  Principles  of  Arches 
of  Bridges,  that  a parabolic  arch  is  an  arch  of  equilibration, 
when  its  extrados,  or  form  of  its  exterior  covering,  is  the 
very  same  parabola  as  the  lower  or  inside  curve.  Hence  then 
a parabolic  arch,  both  for  the  inside  and  outer  form,  will  be 

very 


444 


THEORY  AND  PRACTICE 


very  proper  for  the  structure  of  a powder  magazine.  For, 
the  inside  parabolic  shape  will  be  very  convenient  as  to  room 
for  stowage:  2dly,  the  exterior  parabola,  every  where  parallel 
to  the  inner  one  will  be  proper  enough  to  carry  off  the  rain 
water  : 3dly,  the  structure  will  be  in  perfect  equilibrium  : 
and  4thly,  the  parabolic  curve  is  easily  constructed,  and  the 
structure  erected 

Put,  as  before,  a = kd,  h — dq,  JK. 

b — aq,  x = dp,  and  y = pc  or  ri. 

Then,  by  the  nature  of  the  parabola 


ADC,  b2  : y2 
_ 2hyy 


h 


hy 


X — 


bb 


X — — ■ hence 
, and  i=^~,by  making  y 


constant.  Then  ci  = 


26o 

q.  is  = = a constant  quan- 

66 


tity  = a,  what  it  is  at  the  vertax  ; that  is,  ci  is  every  where 
equal  to  kd. 

Consequently  kr  is  = dp  ; and  since  ri  is  = pc,  it  is  evi- 
dent that  ki  is  the  same  parab*  lie  curve  with  dc,  and  may 
be  placed  any  height  above  it,  always  producing  an  arch  of 
equilibration,  and  very  commodious  for  powder  magazines. 


THEORY  AND  PRACTICE  OF  GUNNERY. 


In  the  Doctrine  of  Motion,  Forces,  &c.  have  been  given 
several  particulars  relating  to  this  subject.  Thus,  in  props. 
19,  20.  21,  22,  is  given  all  that  relates  to  the  parabolic  theory 
of  projectiles,  that  is,  the  mathematical  principles  which 
would  take  place  and  regulate  such  projects  if  they  were  not 
impeded  and  disturbed  in  their  motions  by  the  air  in  which 
they  move.  But  from  the  enormous  resistance  of  that  me- 
dium, it  happens,  that  many'  military  projectiles,  especially' 
the  smaller  balls  discharged  with  the  higher  velocities,  do 
not  range  so  far  as  a 20th  part  of  what  they  would  Daturally 
do  in  empty  space  ! That  theory  therefore  can  only  be  use- 
ful in  some  few  cases,  such  as  in  the  slower  kind  of  motions, 
not  above  the  velocities  of  2,  3,  or  400  feet  per  second,  when 
the  path  of  the  projectile  differs  but  little  perhaps  from  the 
curve  of  a parabola. 

Again, at  art  104,&c.  of  same  doctrine,  are  given  several  other 
practical  rules  and  calculations,  depending  partly  on  the  fore- 
going 


OF  GUNNERY. 


44& 


going  parabolic  theory,  anil  partly  on  the  results  of  certain 
experiments  performed  with  cannon  balls. 

Again,  in  prop  58,  Statics,  are  delivered  the  theory  and 
calculations  of  a beautiful  military  experiment,  invented  by 
Mr  Robins,  for  determining  the  true  degree  of  velocity  with 
which  balls  are  projected  from  guns,  with  any  charges  of 
powder.  The  idea  of  this  experiment,  is  simply,  that  the 
ball  is  discharged  into  a very  large  but  moveable  block  of 
wood,  whose  small  velocity,  in  consequence  of  that  blow,  can 
be  easily  observed  and  accurately  measured  Then,  frem 
this  small  velocity,  thus  obtained,  the  great  one  of  the  ball  is 
immediately  derived  by  this  simple  proportion,  viz.  as  the 
weight  of  the  ball,  is  to  the  sum  of  the  weights  of  the  ball 
and  ttie  block,  so  is  the  observed  velocity  of  the  last,  to  a 4th 
proportional,  which  is  the  velocity  of  the  ball  sought. — It  is 
evident  that  this  simple  mode  of  experiment  will  be  the  source 
of  numerous  useful  principles  as  results  derived  from  the 
experiments  thus  made,  with  all  lengths  and  sizes  of  guns, 
with  all  kinds  and  sizes  of  balls  and  other  shot,  and  with  all 
the  various  sorts  and  quantities  of  gunpowder  ; in  short,  the 
experiment  will  supply  answers  to  all  enquiries  in  projectiles, 
excepting  the  extent  of  their  ranges  ; lor  it  will  even  de- 
termine the  resistance  of  the  air,  by  causing  the  ball  to  strike 
the  block  of  wood  at  different  distances  from  the  gun,  thus 
showing  the  velocity  lost  by  passing  through  those  different 
spaces  of  air  ; all  which  circumstances  are  partly  shown  in 
my  4to  vol  of  Tracts  published  in  1786,  and  which  will  be 
completed  in  my  new  volumes  of  miscellaneous  tracts  now 
printing. 

Lastly,  in  prob.  17,  Prac.  Ex.  on  Forces,  some  results  of  (he 
same  kind  of  experiment  are  successfully  applied  to  deter- 
mine the  curious  circumstances  of  the  first  force  or  elasticity’ 
of  the  air  resulting  from  fired  gunpowder,  and  (he  velocity 
with  which  it  expands  itself.  These  are  circumstances  which 
have  never  before  been  determined  with  any  precision.  Mr. 
Robins,  and  other  authors,  it  may  be  said,  have  only  guessed  at, 
rather  than  determined  them.  That  ingenious  philosopher,  by 
a simple  experiment,  truly  showed  that  by  the  firing  of  a par- 
cel of  gunpowder,  a quantity  of  elastic  air  was  disengaged, 
which,  when  confined  in  the  space  only  occupied  by  the  pow- 
der before  it  was  fired,  was  found  to  be  near  250  times  strong- 
er than  the  weight  or  elasticity  of  the  common  atmospheric  air. 
He  then  heated  the  same  parcel  of  air  to  the  degree  of  red  hot, 
iron,  and  found  it  in  that  temperature  to  be  about  4 times  as 
strong  as  before  ; whence  he  inferred,  that  the  first  strength  of 
the  inflamed  fluid,  must  be  nearly  1000  times  the  pressure  of 

the 


446 


THEORY  AND  PRACTICE 


the  atmosphere.  But  this  was  merely  guessing  at  the  degree 
of  heat  in  the  inflamed  fluid,  and  consequently  of  its  first 
strength,  both  which  in  fact  are  found  to  he  much  greater.  It 
is  true  that  this  assumed  degree  of  strength  accorded  pretty 
well  with  that  author’s  experiments  ; but  this  seeming  a- 
greement,  it  may  easily  be  shown,  could  only  be  owing  to 
the  inaccuracy  of  his  own  further  experiments  ; and.  in  fact, 
with  far  better  opportunities  then  fell  to  the  lot  of  Mr  tob- 
ins,  we  have  shown  that  inflamed  gunpowder  is  about  dou- 
ble the  strength  that  he  has  assigned  to  it,  and  that  it  ex- 
pands itself  with  the  velocity  of  about  5000  feet  per  se- 
cond. 

Fully  sensible  of  the  importance  of  experiments  cf  this 
kind,  first  practised  by  Mr  Robins  with  musket  balls  only, 
my  endeavours  for  many  years  were  directed  to  the  prosecu- 
tion of  the  same,  on  a larger  scale,  with  cannon  balls  and  I 
having  had  the  honour  to  be  called  on  to  give  my  assistance  at 
several  courses  of  such  experiments,  carried  on  at  Woolwich 
by  the  ingenious  officers  of  the  Royal  Artillery  there,  under 
the  auspices  of  the  Masters  General  of  the  Ordnance,  i have 
assiduously  attended  them  for  many  years  The  first  of  these 
courses  was  performed  in  the  year  1775,  being  2 years  after 
my  establishment  in  the  Royal  Academy  at  that  place  . and  in 
the  Philos.  Trans,  for  the  year  1778  I gave  an  account  of 
these  experiments,  with  deductions,  in  a memoir,  which  was 
honoured  with  the  Royal  Society’s  gold  medal  of  that  year. 

In  conclusion,  from  the  whole,  the  following  important  deduc- 
tions were  fairly  drawn  and  stated,  viz. 

1st,  It  is  made  evident  by  these  experiments,  that  gun- 
powder fires  almost  instantaneously.  2 dly,  The  velocities  \ 
communicated  to  shot  of  the  same  weight,  with  • different 
charges  of  powder,  are  nearly  as  the  square  roots  of  those 
charges.  3 dly,  And  when  shot  of  different  weights  are  fired 
with  the  same  charge  of  powder,  the  velocities  communicated 
to  them,  are  nearly  in  the  inverse  ratio  of  the  square  roots 
of  their  weights.  4ihly,  So  that,  in  general,  shot  which  are 
of  different  weights,  and  impelled  by  the  firing  of  different 
charges  of  powder,  acquire  velocities  which  are  directly  as 
the  square  roots  of  the  charges  of  powder,  and  inversely  as 
the  square  roots  of  the  weights  of  the  shot,  bthly,  It  would 
therefore  be  a great  improvement  in  artillery,  occasionally  to 
make  use  of  shot  of  a long  shape,  or  of  heavier  matter,  as 
lead  ; for  thus  the  momentum  of  a shot,  when  discharged 
with  the  same  charge  of  powder,’  would  be  increased  in  the 
ratio  of  the  square  root  of  the  weight  of  the  shot ; which 
w ould  both  augment  proportionally  the  force  of  the  blow  with 

which 


OF  GUNNERY. 


44t 


which  it  would  strike,  and  the  extent  of  the  range  to  which 
it  would  go.  6thly,  It  would  also  be  an  improvement,  to 
diminish  the  windage  ; since  by  this  means,  one  third  or  more 
of  the  quantity  of  powder  might  be  saved.  7 thly,  When  the 
improvements  mentioned  in  the  last  two  articles  are  consider- 
ed as  both  taking  place,  it  appears  that  about  half  the  quantity 
.of  powder  might  be  saved.  But,  important  as  the  saving  may 
be,  it  appears  to  be  still  exceeded  by  that  of  the  guns  : for 
thus  a small  gun  may  be  made  to  have  the  effect  and  execution 
of  another  of  two  or  three  times  its  size  in  the  present  way, 
by  discharging  a long  shot  of  2 or  3 times  the  weight  of  its 
usual  ball,  or  round  shot ; and  thus  a small  ship  might  em- 
ploy shot  as  heavy  as  those  of  the  largest  now  in  use 

Finally,  as  these  experiments  prove  the  regulations  with 
respect  to  the  weight  of  powder  and  shot,  when  discharged 
from  the  same  piece  of  ordnance  ; so,  by  making  similar  ex- 
periments with  a gun  varied  in  its  length  by  cutting  off  from 
it  a certain  part,  before  each  set  of  trials,  the  effects  and  ge- 
neral rules  for  the  different  lengths  of  guns,  may  be  with 
certainty  determined  by  them.  In  short,  the  principles  on 
which  these  experiments  were  made,  are  so  fruitful  in  con- 
sequences, that,  in  conjunction  with  the  effects  of  the  resist- 
ance of  the  medium,  they  appear  to  be  sufficient  for  answering 
all  the  inquiries  of  the  speculative  philosopher,  as  well  as 
those  of  the  practical  artillerist. 

Such  then  was  the  summary  conclusion  from  the  first  set 
of  experiments  with  cannon  balls,  in  the  year  1775,  and  such 
were  the  probable  advantages  to  be  derived  from  them.  I am 
not  aware  however  that  any  alterations  were  adopted  from 
them  by  authority  in  the  public  service  : unless  we  are  to  ex- 
cept the  instance  of  carronades,  a species  of  ordnance  that  was 
afterwards  invented,  and  in  some  degree  adopted  in  the  public 
service  ; for,  in  this  instance,  the  proprietors  of  those  pieces 
by  availing  themselves  of  the  circumstances  of  large  balls,  and 
very  small  windage,  have,  with  small  charges  of  powder,  and 
at  little  expense,  been  enabled  to  produce  very  considerable 
and  useful  effects  with  those  light  pieces. 

The  2d  set  of  these  experiments  extended  through  most 
part  of  the  summer  seasons  of  the  years  1783,  1784,  1785, 
and  some  in  1786.  The  objects  of  this  course  were  nume- 
rous and  various  : but  the  principal  articles  as  follow  : 1.  The 
velocities,  with  which  balls  are  projected  by  equal  charges  of 
powder,  from  pieces  of  equal  weight  and  calibre,  but  of  dif- 
ferent lengths.  2.  The  velocities  with  different  charges  of 
powder,  the  weight  and  length  of  the  guns  being  equal. 
3.  The  greatest  velocities  due  to  the  different  lengths  of  guns, 

to 


■448 


THEORY  AND  PRACTICE 


to  be  ascertained  by  successively  increasing  the  charge,  till 
the  bore  should  be  filled,  or  till  the  velocity  should  decrease 
again.  4.  The  effect  of  varying  the  weight  of  the  piece  ; 
every  thing  else  being  tie  same.  5 lie  penei  rations  of 
ball?  into  blocks  of  wood  6 The  ranges  and  times  of  flight 
of  balls  ; to  compare  them  with  their  first  velocities,  for  ascer- 
taining the  resistance  of  the  medium.  7.  The  effect  of  wads  ; 
of  different  degrees  of  ramming,  or  compressing  the  charge  ; 
of  different  degrees  of  windage  ; of  different  positions  of  the 
vent  ; of  chambers  and  trunnions,  and  every  other  cir- 
cumstance necessary  to  be  known  for  the  improvement  of  ar- 
tillery. 

An  ample  account  is  given  of  these  experiments,  and  the 
results  deduced  from  them  in  my  volume  of  Tracts  published 
in  1786  ; some  few  circumstances  only  of  which  can  he  noted 
here.  In  this  course,  4 brass  guns  were  employed,  very 
nicely  bored  and  cast  on  purpose,  of  different  lengths,  but 
equal  in  all  other  respects,  viz.  in  weight  and  bore,  &c.  The 
lengths  of  the  bores  of  the  guns  were, 

the  gun  n°  1,  was  15  calibres,  length  of  bore  28-5  inc. 

. . . n°  2,  . 20  calibres,  ....  3fi-4 

. n°  3,  . 30  calibres,  ....  57*7 

. . . n°  4,  . 40  calibres,  ....  80  2. 

the  calibre  of  each  being  2T'7  inches,  and  the  medium  weight 
of  the  balls  16  oz.  13  drams. 

The  mediums  of  all  the  experimented  velocities  of  the 
balls,  with  which  they  struck  the  pendulous  block  of  wood,  . 
placed  at  the  distance  of  32  feet  from  the  muzzle  of  the  gun, 
for  several  charges  of  powder,  were  as  in  the  following  table. 


Table  of  Initial  Velocities 

1'ovvder. 

The  Guns. 

oz. 

No.  1 

'•o.  2 

No.  3 

No  4 

2 

780 

835 

920 

970 

4 

1 100 

) 180 

1300 

1370 

6 

1340 

1445 

1590 

1680 

8 

1430 

1 580 

1790 

1940 

12 

1 436 

16  40 

. 

14 

1660 

16 

. 

2000 

# 

18 

• 

• 

• 

2200 

placed  in  the  1st  column,  for  all  the  four  guDS,  the  num- 
bers denoting  so  many  feet  per  second.  Whence  in  general 

it 


OF  GUNNERY. 


449 


it  appears  how  the  velocities  increase  with  the  charges  of 
powder,  for  each  gun,  and  also  how  they  increase  as  the  guns 
are  longer,  with  the  same  charge,  in  every  instance. 

By  increasing  the  quantity  of  the  charges  continually,  for 
each  gun,  it  was  found  that  the  velocities  continued  to  increase 
till  they  arrived  at  a certain  degree,  different  in  each  gun  ; after 
which,  they  constantly  decreased  again,  till  the  bore  was  quite 
tilled  with  the  charge.  The  charges  of  powder  when  the 
velocities  arrived  at  their  maximum  or  greatest  state,  were 
various,  as  might  be  expected,  according  to  the  lengths  of  the 
guns  ; and  the  weight  of  powder,  with  the  length  it  extended 
in  the  bore,  and  the  fractional  part  of  the  bore  it  occupied, 
are  shown  in  the  following  table,  of  the  charges  for  the  great- 
est effect. 


Gun, 

n°. 

Length 
of  the 
Bore 

The  Charge. 

Weight, 

oz. 

Len 

gth. 

Inches. 

Part  of 
whole 

1 

28-5 

12 

8-2 

_3_ 

2 

38-4 

14 

9-5 

3 

1 2 

3 

57-7 

16 

10-7 

3 

1 6 

4 

80-2 

18 

121 

3 

2 0 

Some  few  experiments  in  this  course  were  made  to  obtain 
the  ranges  and  times  of  flight,  the  mediums  of  which  are  ex- 
hibited in  the  following  table. 


Guns 

Pow- 

der 

Bal 

Weight. 

Is. 

Diam. 

Elevat 

gun. 

Time 
of  flight. 

Range. 

First 

veloc. 

oz. 

oz. 

dr. 

inch. 

secs. 

feet. 

feet. 

n°2. 

2 

16 

10 

1-96 

45° 

21-2 

5109 

863 

do. 

2 

16 

5 

1-96 

15 

9-2 

4130 

868 

do. 

4 

16 

8 

1-96 

15 

9-2 

4660 

1234 

do. 

8 

16 

12 

1-96 

15 

14-4 

6066 

1644 

do. 

12 

16 

12 

1-95 

15 

15-5 

6700 

1676 

n°3. 

8 

15 

8 

1-96 

15 

10-1 

5610 

1938 

In  this  table  are  contained  the  following  concomitant  data, 
determined  with  a tolerable  degree  of  precision  ; viz.  the 
weight  of  the  powder,  the  weight  and  diameter  of  the  ball, 
the  initial  or  projectile  velocity,  the  angle  ©f  elevation  of  the 
Vol.  II.  58  guts, 


450 


THEORY  AND  PRACTICE 


gun,  the  time  in  seconds  of  the  ball’s  flight  through  the  air 
and  its  range,  or  the  distance  where  it  fell  on  the  horizontal 
plane.  From  which  it  is  hoped  that  some  aid  may  be  derived 
towards  ascertaining  the  resistance  of  the  medium,  and  its 
effects  on  other  elevations,  &c.  and  so  afford  some  means  of 
obtaining  easy  rules  for  the  cases  of  practical  gunnery. 
Though  the  completion  of  this  enquiry,  for  want  of  time  at 
present,  must  be  referred  to  another  work,  where  we  may 
have  an  opportunity  of  describing  another  more  extended 
course  of  experiments  on  this  subject,  which  have  never  yet 
been  given  to  the  public. 

Another  subject  of  enquiry 
in  the  foregoing  experiments, 
was,  how  far  the  halls  would 
penetrate  into  solid  blocks  of 
elm  wood,  fired  in  the  direc- 
tion of  the  fibres.  The  an- 
nexed tablet  shows  the  results 
of  a few  of  the  trials  that 
were  made  with  the  gun  n°  2, 
with  the  most  frequent  charges 
of  2,  4,  and  8 ounces  of  pow- 
der ; and  the  mediums  of  the 
penetrations,  as  placed  in  the 
last  line,  are  found  to  be  7,  15, 
and  20  inches,  with  those  charges.  These  penetrations  are 
nearly  as  the  numbers 

2,  4,  6,  or  1,  2,  3 ; but  the  charges  of  powder  are  as 

2,  4,  8,  or  1,  2,  4 ; so  that  the  penetrations  are  propor- 
tional to  the  charges  as  far  as  to  4 ounces,  but  in  a less  ratio 
at  8 ounces  ; whereas,  by  the  theory  of  penetrations  the 
depths  ought  to  be  proportional  to  the  charges,  or  which  is 
the  same  thing,  as  the  squares  of  the  velocities.  So  that  it 
seems  the  resisting  force  of  the  wood  is  not  uniformly  or  con- 
stantly the  same  but  that  it  increases  a little  with  the  increased 
velocity  of  the  hall.  This  may  probably  be  occasioned  by 
the  greater  quantity  of  fibres  driven  before  the  ball  ; which 
may  thus  increase  the  spring  and  resistance  of  the  wood,  and 
prevent  the  ball  from  penetrating  so  deep  as  it  otherwise 
might  do. 

From  a general  inspection  of  this  second  course  of  these 
experiments,  it  appears  that  all  the  deductions  and  observa- 
tions made  on  the  former  course,  are  here  corroborated  and 
strengthened,  respecting  the  velocities  and  weights  of  the 
halls,  and  charges  of  powder,  &c.  It  further  appears  also 
that  the  velocity  of  the  ball  increases  with  the  increase  of 

charge 


Penetrations  of  Balls  into 

solid 

Elm  wood. 

Powder  2 

4 

8 oz. 

7 

16-6 

18-9 

13-5 

21-2 

18  1 

20-8 

20-5 

Means  7 

15 

20 

OF  GUNNERY. 


451 


charge  only  to  a certain  point,  which  is  peculiar  to  each  gun, 
where  it  is  greatest  : and  that  by  further  increasing  the 
charge,  the  velocity  gradually  diminishes,  till  the  bore  is 
quite  full  of  powder.  That  this  charge  for  the  greatest  ve- 
locity is  greater  as  the  gun  is  longer,  but  yet  not  greater  in 
so  high  a proportion  as  the  length  of  the  gun  is  ; so  that  the 
part  of  the  bore  filled  with  powder,  bears  a less  proportion  to 
the  whole  bore  in  the  long  guns,  than  it  does  in  the  shorter 
ones  ; the  part  which  is  filled  being  indeed  nearly  in  the  in- 
verse ratio  of  the  square  root  of  the  empty  part. 

It  appears  that  the  velocity,  with  equal  charges,  always 
increases  as  the  gun  is  longer  ; though  the  increase  in  velo- 
city is  but  very  small  in  comparison  to  the  increase  in  length  ; 
the  velocities  being  in  a ratio  somewhat  less  than  that  of  the 
square  roots  of  the  length  of  the  bore,  but  greater  than  that 
of  the  cube  roots  of  the  same,  and  is  indeed  nearly  in  the  mid- 
dle ratio  between  the  two. 

It  appears  from  the  table  of  ranges,  that  the  range  increa- 
ses in  a much  lower  ratio  than  the  velocity,  the  gun  and  ele- 
vation being  the  same.  And  when  this  is  compared  with  the 
proportion  of  the  velocity  and  length  of  gunj  in  the  last  para- 
graph, it  is  evident  that  we  gain  extremely  little  in  the  range 
by  a great  increase  in  the  length  of  the  gun,  with  the  same 
charge  of  powder.  In  fact  the  range  is  nearly  as  the  5th  root 
of  the  length  of  the  bore  : which  is  so  small  an  increase,  as 
to  amount  only  to  about  a 7th  part  more  range  for  a double 
length  of  gun. — From  the  same  table  it  also  appears,  that  the 
time  of  the  ball’s  flight  is  nearly  as  the  range  ; the  gun  and 
elevation  being  the  same. 

It  has  been  found,  by  these  experiments,  that  no  difference 
2s  caused  in  the  velocity,  or  range,  by  varjing  the  weight  of 
the  gun,  nor  by  the  use  of  wads,  nor  by  different  degrees  of 
ramming,  nor  by  firing  the  charge  of  powder  in  different 
parts  of  it.  But  that  a very  great  difference  in  the  velocity 
arises  from  a small  degree  in  the  windage  : indeed  with  the 
usual  established  rvindage  only,  viz.  about  ^ of  the  calibre, 
no  less  than  betwen  i and  i of  the  powder  escapes  and  is 
lost:  and  as  the  balls  are  often  smaller  than  the  regulated 
size,  it  frequently  happens  that  half  the  powder  is  lost  by  un- 
necessary windage. 

It  appears  too  that  the  resisting  force  of  wood,  to  balls 
fired  into  it,  is  not  constant  : and  that  the  depths  penetrated 
by  balls,  with  different  velocities  or  charges,  are  nearly  as  the 
logarithms  of  the  charges,  instead  of  being  as  the  charges 
themselves,  or,  which  is  the  same  thing,  as  the  square  of  the 
velocity. — Lastly,  these  and  most  other  experiments,  show, 

that 


452 


THEORY  AND  PRACTICE 


that  balls  are  greatly  deflected  from  the  direction  in  which 
they  are  projected  ; and  that  as  much  as  300  or  40©  yards  in 
a range  of  a mile,  or  almost  |th  of  the  range. 

We  have  before  adverted  to  a third  set  of  experiments,  of 
still  more  importance,  with  respect  to  the  resistance  of  the 
medium,  than  any  of  the  former  ; but,  till  the  publication  of 
those  experiments,  we  cannot  avail  ourselves  of  all  the  disco- 
veries they  contain  In  the  mean  time  however  we  may  ex- 
tract from  them  the  three  follow  ing  tables  of  resistances,  for 
three  different  sizes  of  balls,  and  for  velocities  between  IOC- 
feet  and  200©  feet  per  second  of  time. 


Table 

I, 

Table  1J. 

Table  II I. 

Resistances  to  a 

ball  of  1-965 

Resistances  to  a 

Resist,  to  a 

bull 

mciits  diameter,  and  16  oz. 

oall  2'78  in.  diam ■ 

3 '55  772-  diam.  and 

13  dr.  weight. 

and  3 Lb.  weight. 

6lb-  1 oz-  8 dr.  wt. 

Vel 

Resistances. 

1 Dif 

2dD>f. 

Vel- 

Res 

D.ts 

Vel. 

Res. 

Difs, 

fe“t 

lbs 

ozs. 

feet 

lbs 

fee  t 

lbs. 

too 

0 :7 

25 

8j 

14 

20 

27 

35 

44 

54 

66 

79 

96 

900 

35 

5 

1700 

115 

9 

200 

0 69 

n 

55 

950 

41 

1-50 

124 

Q 

.-•'00 

i56 

25 

6 

1000 

47 

1300 

133 

Q 

*00 

2 81 

45 

7 

lO-.O 

53 

7 ! 

!1350 

142 

10 

10 

104 

lii 

13 

14 

600 

4 50 

72 

8 

1100 

60 

7 

1400 

152 

600 

6 69 

107 

9 

1150 

67 

1450 

162 

70u 

800 

9 44 
12  81 

151 
: 05 

10 

12 

1200 

1250 

7 4 
82 

8 

1500 

1550 

172$ 

184 

900 

16  94 

271 

13 

j 1300 

91 

10 

11 

;03 

10 

1600  197 

1000 

2r88 

350 

IS 

13^0 

101 

1650 

211 

15 

16 
17 

100 

27-63 

442 

164 

115 

124 

131 

135 

135 

12 

1400 

112 

1700  226 

1*00 

34  13 

546 

11 

1450 

122  A 

1750 

242 

1 -00 

41  31 

661 

9 

1500 

13?i 

1800 

259 

1400 

490- 

785 

7 

1550 

141$ 

H 

8 

1500 

57  75 

916 

4 

1600 

150 

1.  CO 

65  69 

.051 

0 

1650 

158 

- 

1700 

74-13 

' 1 86 

-2 

17.  0 

165 

l 

1800 

<52-44 

.319 

1 JO 
128 
122 

—5 

1750 

171 

1900 

90  44 

.447 

-6 

176 

0 

2006 

yb-ofa 

1569 

j 1800 

1 

. 

PROBLEM  I. 

To  determine  the  Resistance  of  the  Medium  against  a Rail 
of  any  other  size,  moving  with  any  of  the  Velocities  given  in  the 
foregoing  Tables. 

The  analogy  among  the  numbers  in  all  these  tables  is 
very  remarkable  and  uniform,  the  same  general  law’s  running 

through 


GP  GUNNERY. 


453 


through  them  all.  The  same  laws  are  also  observable  as  in 
the  table  of  resistances  in  page  412  of  this  volume,  parti- 
cularly the  1st  and  2d  remarks  immediately  following  that 
table,  viz.  that  the  resistances  increase  in  a higher  proportion 
than  the  square  of  the  velocities,  with  the  same  body  ; and 
that  the  resistances  also  increase  in  a rather  higher  ratio  than 
the  surfaces,  with  different  bodies,  but  the  same  velocity.  Yet 
this  latter  case,  viz  the  ratios  of  the  resistances  and  of  the 
surfaces,  or  of  the  squares  of  the  diameters  which  is  the  same 
thing,  are  so  nearly  alike,  that  they  may  be  considered  as 
equal  to  each  other  in  any  calculations  relating  to  artillery  prac- 
tice. Fcr  example,  suppose  it  were  required  to  determine 
what  would  be  the  resistance  of  the  air  against  a 24lb  ball  dis- 
charged with  a velocity  of  2000  feet  per  second  of  time. 
Now,  by  the  1st  of  the  foregoing  tables,  the  ball  of  1*965 
inches  diameter,  when  moving  with  the  velocity  2000,  suffer- 
; ed  a resistance  of  981b  : then  since  the  resistances,  with  the 
' same  velocity,  are  as  the  surfaces  ; and  the  surfaces  are  as 
the  squares  of  the  diameters  ; and  the  diameters  being  T965 
and  5-6.  the  squares  of  which  are  3-86  and  31  36,  therefore  as 
3 86  : 31-36  : : 981b  : 7961b  ; that  is,  the  24lb  ball  would  suf- 
fer the  enormous  resistance  of  7961b  in  its  flight,  in  opposition 
to  the  direction  of  its  motion  ! 

And.  in  general,  if  the  diameter  of  any  proposed  ball  be 
denoted  by  d,  and  r denote  the  resistance  in  the  1st  table 

Cl2  p 

due  to  the  proposed  velocity  of  the  1‘965  ball  ; then  will 

denote  the  resistance  with  the  same  velocity  against  the  ball 
whose  diameter  is  d ; or  it  is  nearly  id2 r which  is  but  the  28th 
part  greater  than  the  former. 

PROBLEM  II. 

To  assign  a Rule  for  determining  the  Resistance  due  to  any 
Indeterminate  Velocity  of  a Given  Ball. 

This  problem  is  very  difficult  to  perform  near  the  truth, 
on  account  of  the  variable  ratio  which  the  resistance  bears  to 
the  velocity,  increasing  always  more  and  more  above  that  of 
the  square  of  the  velocity,  at  least  to  a certain  extent  ; and 
indeed  it  appears  that  there  is  no  single  integral  power  what- 
ever of  the  velocity,  or  no  expression  of  the  velocity  in  one 
term  only,  that  can  be  proportional  to  the  resistances  through- 
out. It  is  true  indeed  that  such  an  expression  can  be  assigned 
by  means  of  a fractional  power  of  the  velocity,  or  rather  one 

whose  index  is  a mixed  number,  viz.  2 or  2-1  ; thus  — — = 

the 


454 


THEORY  AND  PRACTICE 


the  resistance,  is  a formula  in  one  term  only,  which  will 
answer  to  all  the  numbers  in  the  first  table  of  resistances  very 
nearly,  and  consequently,  by  means  of  the  ratio  of  the  squares 
of  the  diameters  of  the  balls,  for  any  other  balls  whatever. 
This  formula  then,  though  serving  quite  well  for  some  par- 
ticular resistance,  or  even  for  constructing  a complete  series 
or  table  of  resistances,  is  not  proper  for  the  use  of  problems 
in  which  fluxions  and  fluents  are  concerned,  on  account  of  the 
mixed  number  2Tr7,  in  the  index  of  the  velocity  v. 

We  must  therefore  have  recourse  to  an  expression  in  two 
terms,  or  a formula  containing  two  integral  powers  of  the 
velocity,  as  v 2 and  v,  the  first  and  2d  powers,  affected  with 
general  coefficients  m and  n,  as  mv2  + nv  — r the  resist- 
ance. Now,  to  determine  the  general  numerical  values  of 
the  coefficients  m and  n , we  must  adapt  this  general  ex- 
pression mv2  -f-  nv  = r,  to  two  particular  cases  of  velocity 
at  a convenient  distance  from  each  other,  in  one  of  the  fore- 
going tables  of  resistances,  as  the  first  for  instance.  Now, 
after  making  several  trials  in  this  way,  1 have  found  that  the 
two  velocities  of  500  and  1000  answer  the  general  purpose 
better  than  any  other  that  has  been  tried.  Thus  then,  em- 
ploying these  two  cases,  we  must  first  make  v — 500,  and 
r = 44lb,  its  correspondent  resistance,  and  then  again  v = 
1000,  and  r = 21881b,  the  resistance  belonging  to  it ; this 
will  give  two  equations  by  which  the  general  value  of  m and 
©f  n will  be  determined.  Thus  then  the  two  equations  being 
5002m  -j-  500n  = 4-5, 
and  10002m  -j-  lOOOn.  = 2T88  ; 
dividing  the  1st  by  500,  and  the  < 500m  + n = -009, 

2d  by  1000,  they  are  . . ( 1000m  + n = '02188  ; 

the  dif.  of  these  is 500m  = 01288, 

and  therefore  div.  by  500,  gives  m = -00002576  ; 
hence  n — -009  — 500m  = '009  — -01288  — — -00388  = n. 
Hence  then  the  general  formula  will  be  00002576u2  — 
00388  v = rthe  resistance  nearly  in  avoirdupois  pounds,  in  all 
cases  or  all  velocities  whatever. 


Now, 


OF  GUNNERY. 


Now,  to  find  how  near  to  the 
truth  this  theorem  comes,  in 
every  instance  in  the  table,  by 
substituting  for  v,  in  this  formula, 
all  the  several  velocities,  100, 

200,  300,  &c.  to  2000,  these  give 
the  correspondent  values  of  r,  or 
the  resistances,  as  in  the  2d  co- 
lumn of  the  annexed  table,  their 
velocities  being  in  the  first  co- 
lumn ; and  the  real  experimented 
i resistances  are  set  opposite  to 
them  in  the  3d  or  last  column  of 
I the  same.  By  the  comparison  of 
the  numbers  in  these  two  co- 
lumns together,  it  is  seen  that 
there  are  no  where  any  great  dif- 
ference between  them,  being 
sometimes  a little  in  excess,  and 
again  a little  in  defect,  by  very 
small  differences  ; so  that,  on  the 
whole,  they  will  nearly  balance 
one  another,  in  any  particular  in- 
stance of  the  range  or  flight  of 
a ball,  in  all  degrees  of  its  velocity,  from  the  first  or 
greatest,  to  the  smallest  or  last.  Except  in  the  first  two  or 
three  numbers,  at  the  beginning  of  the  table,  for  the  veloci- 
ties 100,  200,  300,  for  which  cases  another  theorem  may  be 
employed.  Now,  in  these  three  velocities,  as  well  as  in  all 
that  are  smaller,  down  to  nothing,  the  theorem  -00001725t>2 
j — r the  resistance,  will  very  well  serve,  as  it  brings  out  for 
the  first  three  resistances  -17  and  69  and  T55i,  differing 
in  the  last  only  by  a very  small  fraction. 

Carol.  1.  The  foregoing  rule  ‘00002576V2  — •00388'u  = r, 
denotes  the  resistance^  for  the  ball  in  the  first  table,  whose 
diameter  is  1-965,  the  square  of  which  is  3-86  or  almost  4 ; 
hence  to  adapt  it  to  a ball  of  any  other  diameter  cl,  we  have 
only  to  alter  the  former  in  proportion  to  the  squares  of  the 

diameters,  by  which  it  becomes  (‘00002576v2 — -003830 

j-o6 

• = (00000667t>2  — -001o)d2  = (-00000|-o2  — -00!o)d2,  which 
is  the  resistance  for  the  ball  whose  diameter,  is  d,  with  the 
velocity  v. 

Corol.  2.  And,  in  a similar  manner,  to  adapt  the  theorem 
0OOO1725%>2  = r,  for  the  smaller  velocities,  to  any  other  size 


4oo 


Velocs. 
or  v. 

Comput 

resists 

Exper. 

resists 

100 

—•13 

•17 

200 

—•25 

•69 

300 

1-15 

1-56 

400 

2-57 

2 81 

500 

4 50 

4-50 

600 

6-94 

6 69 

700 

9-90 

9-44 

800 

13-38 

12-81 , 

900 

17-37 

16-94 

1000 

21-88 

21-88 

1100 

26-90 

27-63 

1200 

32-44 

34- 13 

1300 

38-49 

41-31 

1400 

4506 

49-06 

1500 

5214 

57-25 

1600 

59  74 

65-69 

1700 

67-85 

7413 

1800 

76-48 

82-44 

1900 

85-62 

90-44 

2000 

95-28 

98-06 

456 


THEORY  AND  PRACTICE 


of  ball,  we  must  multipty  it  by  , the  ratio  of  the  surfaces 

0*00 

by  which  it  becomes  •00000447r/2ns  = r. 

We  shall  soon  take  occasion  to  make  some  applications  in 
the  use  of  the  foregoing  formulas,  after  considering  the  effect* 
of  such  velocities  in  the  cases  of  nonresistances. 

PROBLEM  HI. 

To  determine  the  Height  to  which  a Ball  will  rise,  whet.  J 
fired  from  a cannon  Perpendicularly  Upwards  with  a Given  i 
Velocity,  in  a Nonresisting  Medium,  or  supposing  no  Resis- 
tance in  the  Air. 

By  art  73,  Motion  and  Forces,  this  vol  it  appears  that  any 
body  projected  upwards,  with  a given  velocity,  will  ascend  to  : 
the  height  due  to  the  velocity,  or  the  height  from  which  it 
must  naturally  fall  to  acquire  that  velocity  ; and  the  spaces 
fallen  being  as  the  square  of  the  velocities  ; also  16  feet  being 
the  space  due  to  the  velocity  32  ; therefore  the  space  due  to 
any  proposed  velocity  v,  will  be  found  thus,  as  322  : 16  : : v2:  s 
the  space  or  as  64  : 1 : : v2  : v2  =s  the  space,  or  the  height 

to  which  the  velocity  v will  cause  the  body  to  rise  independ- 
ent of  the  air’s  resistance. 

Exam.  For  example,  if  the  first  or  projectile  velocity,  be 
20U0  feet  per  second,  being  nearly  the  greatest  experimented 
velocity,  then  the  rule  f^v2  ■=  s becomes  X 200Q2  = 62500 
feet  = Ilf  miles  : that  is,  any  body,  projected  with  the  ve- 
locity 2000  feet,  would  ascend  nearly  12  miles  in  height,  with- 
out resistance. 

Corol.  Because,  by  art.  88  Projectiles  this  vol.  the  greatest 
range  is  just  double  the  height  due  to  the  projectile  velocity, 
therefore  the  range  at  an  elevation  of  45°,  with  the  velocity 
in  the  last  example,  would  be  23|  miles  in  a nonresisting  me- 
dium. We  shall  now  see  what  the  effects  will  be  with  the 
resistance  of  the  air. 

PROBLEM  IV. 

To  determine  the  Height  to  which  a Ball  projected  Upwards, 
as  in  the  last  problem,  will  ascend,  being  Resisted  by  the  Atmos- 
phere. 

Putting  x to  denote  any  variable  and  increasing  height  as- 
cended by  the  ball;  v its  variable  and  decreasing  velocity  there; 
d the  diameter  of  the  ball,  its  weight  being  w ; m — -OOOOOf , 
and  n = *001,  the  co-etficients  of  the  two  terms  denoting  the 
law  of  the  air’s  resistance.  Then  (mi>2  — nv)d2 , by  cor.  1 to 


OF  GUNNERY. 


457 


prob.  2,  will  be  the  resistance  of  the  air  against  the  ball  in 
avoirdupois  pounds  : to  which  if  the  weight  of  the  ball  be  add- 
ed, then  ( mv 2 — nv)d 2 -f-  w will  be  the  whole  resistance  to  the 
ball’s  motion  ; this  divided  by  w,  the  weight  of  the  ball  in 

motion,  gives  - == d2-f  1 = f the  retard- 

ww 

ing  force.  Hence  the  general  formula  vy  = 2 gfx  (theor.  10 

pa.  379  this  volume)  becomes  — vy  = 2 gx  X 

making  y negative  because  v is  decreasing,  where  g — Id  ft.  ; 
and  hence 


x — — — X — , = 

2g  (mv2 — -j-iv 


2gnid2 


» . IV 

1)2 V -] — 

m ma* 


Now,  for  the  easier  finding  the  fluent  of  this,  assume 


v — — = z ; then  v = z -j and  v2 

2 m 


2m 

and  vv  = z'z  -f- 

n2 


+ — * + 


n2 
4m 3 


2 m 


•z , and  v2 


n . n2  „ „ 

— v 4 = z2,  and  v2 

m 4m2 


v — z2  — — — ; these  being  substituted  in  the  above  value 
m 4 m2  ° 

of  x,  it  becomes  x — 


2 gmd.2 


X. 


. n. 

2Z  + — Z 
2 m 


n2  w 

4m2  md 3 


2gmd"  ^ 


__  y Zz4pi 

2 2 gmd2  z2iq2 

1 md 2 ^ 


...  71  , 0 W 0 „ , 0 W 

PuttlDs  J5  = 2^’ and  ? = ^r-R“’or/,“  + r = —■ 

Then  the  general  fluents,  taken  by  the  8th  and  11th  forms 


of  the  table  of  Fluents  .give  x~ 


2 gmd 


X[AlOg.  (z2  + ?2)  -f-  r-X 


P_ 

q2 


arc  to  rad.  q and  tan.  zl  = - — ^-X  |\Gog.  (v2 — % -f-  — ) -f- 

7 1 2 gmd2  12  a ' m mu 2 7 

~ X arc  to  rad.  q and  tang,  v — p~\.  But,  at  the  beginning 

of  the  motion,  when  the  first  velocity  is  v for  instance,  and  the 
space  x is  = 0,  this  fluent  becomes 


0 = — — - X [1  log  (v3 

2 gmd2  12  s K 


■ — v -f-  — -)  -f-  ~ X arc  radius  q 
m md 2 q-  7 


tan.  v — p.]  Hence  by  subtraction,  and  taking  v — 0 for 
the  end  of  the  motion,  the  correct  fluent  becomes 

x = — X[Uog-(v2  — — v + — ) — Hog-  — + — X (arc 

tan.  v — . p — arc  tan.  — p to  rad  9)]. 

But  as  part  of  this  fluent,  denoted  by  X the  dif.  of  the  two 

arcs  to  tans,  v — p and  — - p,  is  always  very  small  in  com- 
Vol.  11.  . 59  parisen 


458 


THEORY  AND  PRACTICE 


parisoa  with  the  other  preceding  terms,  they  may  be  omitted 
without  material  error  in  any  practical  instance  ; and  then  the 

„ n , w 
v2  — . V + — 

fluent  is  x = - W X hyp.  log 1 — , for  the  ut- 

4 gmd*  w 

vid2 

most  height  to  which  the  ball  will  ascend,  when  its  motion 
ceases,  and  is  stopped,  partly  by  its  own  gravity,  but  chiefly 
by  the  resistance  of  the  air. 

But  now,  for  the  numerical  value  of  the  general  coefficient 
*10  10 

, and  the  term  — ; because  the  mass  of  the  ball  to  the 

4 gmd2  mil2 

diameter  d,  is  ‘523 6d3,  if  its  specific  gravity  be  s,  its  weight 

will  be  *5236sd3  =®  ; therefore  — = *5236sd,  and  = 

ds  mil2 


11) 

78540sd,  this  divided  by  4 g or  64  it  gives^— = I227*2sd 

for  the  value  of  the  general  coefficient,  to  any  diameter  d 
and  specific  gravity  s.  And  if  we  further  suppose  the  ball 
to  be  cast  iron,  the  specific  gravity,  or  weight  of  one  cubic 
inch  of  which  is  *26855,  it  becomes  330<2,  for  that  coeffi- 
cient ; also  78540sd  = 21090eZ  = -^-,and-  = 150.  And 

md 2 m 


hence  the  foregoing  fluent  becomes  330d  X hyp.  log. 


v2  — 150v-f  21090c? 
21090a? 


or  760 d X com,  log. 


v2  _ 15«V  + 21090(1 

210904 


changing  the  hyperbolic  for  the  common  logs.  And  this  is 
a general  expression  for  the  altitude  in  feet,  ascended  by  any 
iron  ball,  whose  diameter  is  d inches,  discharged  with  any 
velocity  v feet.  So  that,  substituting  any  values  of  d and  v, 
the  particular  heights  will  be  given  to  which  the  balls  will 
ascend,  which  it  is  evident  will  be  nearly  in  proportion  to  the 
diameter  d. 


Exam.  1.  Suppose  the  ball  be  that  belonging  to  the  first 
table  of  resistances,  its  weight  being  16  oz.  13  dr.  or  1*05  lb, 
and  its  diameter  1*965  inches,  when  discharged  with  the  ve- 
locity 2000  feet,  being  nearly  the  greatest  charge  for  any  iron 
ball.  The  calculation  being  made  with  these  values  of  d and 
v,  the  height  ascended  is  found  to  be  2920  feet,  or  little  more 
than  half  a mile  ; though  found  to  be  almost  12  miles  with- 
out the  air’s  resistance.  And  thus  the  height  may  be  found 
for  any  other  diameter  and  velocity. 

Exam.  2.  Again,  for  the  24  lb  ball,  with  the  same  velo- 
city 2000,  its  diameter  being  5*6  = d.  Here  760 d — 4256  \ 

\ v*  — 150v+  21090c?  S8181  ,,  , c ,.  . . , ,„QtQ 

and — = — — , the  log.  of  which  is  1*50958 

210904  11S1  5 & 

theref. 


OF  GUNNERY. 


459 


theref.  1-50958  X 4256  — 6424  = x the  height,  being  a lit- 
tle more  than  a mile. 

We  may  now  examine  what  will  be  the  height  ascended, 
considering  the  resistance  always  as  the  square  of  the  velocity. 

PROBLEM  V. 

To  determine  the  Height  ascended  by  a Ball  projected  as  in 
the  two  foregoing  problems  ; supposing  the  Resistance  of  the  Air 
to  be  as  the  Square  of  the  Velocity. 

Here  it  will  be  proper  to  commence  with  selecting  some 
expex-imented  resistance  corresponding  to  a medium  kind  of 
velocity  between  the  first  or  greatest  velocity  and  nothing, 
from  which  to  compute  the  other  general  resistances,  by  con- 
i sidering  them  as  the  squares  of  the  velocities.  It  is  proper 
to  assume  a near  medium  velocity  and  its  resistance,  because, 
if  we  assume  or  commence  with  the  greatest,  or  the  velocity 
of  projection,  and  compute  from  it  downwards,  the  resistances 
will  be  every  where  too  great,  and  the  altitude  ascended  much 
[i  less  than  just  ; and,  on  the  other  hand,  if  we  assume  or  com- 
mence with  a small  resistance,  and  compute  from  it  all  the 
others  upwards,  they  will  be  much  too  little,  and  the  com- 
puted altitude  far  too  great.  But,  commencing  with  a me- 
dium degree,  as  for  instance  that  which  has  a resistance 
about  the  half  of  the  first  or  greatest  resistance,  or  rather  a 
little  more,  and  computing  from  that,  then  all  those  computed 
resistances  above  that,  will  be  rather  too  little,  but  all  those 
below  it  too  great  ; by  which  it  will  happen,  that  the  defect  of 
the  one  side  will  be  compensated  by  the  excess  on  the  other, 
and  the  final  conclusion  must  be  near  the  truth. 

Thus  then,  if  we  wish  to  determine,  in  this  way,  the  alti- 
tude ascended  by  the  ball  employed  in  the  1st  table  of  resis- 
tances when  projected  with  2000  feet  velocity  ; we  perceive 
by  the  table,  that  to  the  velocity  2000  corresponds  the  re- 
’ sistance  981b  ; the  half  of  this  is  49  to  which  resistance 
corresponds  the  velocity  1400,  in  the  table,  and  the  next 
greater  velocity  1500,  with  its  resistance  57i,  which  will  be 
properest  to  be  employed  here.  Hence  then,  for  any  other 
velocity  v,  in  general,  it  will  be,  according  to  the  law  of  the 

squares  of  the  velocities,  as  15002  : v2  : : 57J-  : = 

•000025fu2  = a-o- , putting  a = -000025A,  which  will  denote 
the  air’s  resistance  for  any  velocity  v,  very  nearly,  counting 
from  2000. 

Now  let  x denote  the  altitude  ascended  when  the  velocity, 
is  v,  and  aj  the  weight  of  the  ball  : then,  as  above,  an2 , is  the 

resistance 


466 


THEORY  AND  PRACTICE 


resistance  from  the  air,  hence  av2  -f-  w is  the  whole  resisting 
force,  and  ax  =f  the  retarding  force  ; 

therefore  — vi,  = 2gfx  = X 2 gx  ; 


and  hence  x = — — X — 

2g  av  2 w 


2?a 


a 


the  fluent  of  which,  by  form  8,  is  X h.  log.  (i>2+— ) ? 

which  when  = 0,  and  z>  = v the  first  or  projectile  velocity  ; 

becomes  0 = — - X h.  1.  (v2  -j ) ; theref  by  subtracting 

4 pa  J ° 


av  - + w 


the  correct  fluent  is  x = — X h 1. , the  height  x 

4 ga  av-  -f-  w 

when  the  velocity  is  reduced  to  v ; and  when  v = 0,  or  the 

velocity  is  quite  exhausted,  this  becomes  — X h.  1.  - — 

J n i 4 ga  w 

for  the  whole  height  to  which  the  ball  will  ascend. 

Ex.  1.  The  values  of  tbe  letters  being  w = 1 -05lb,  4g=64, 

a — •000025A,  the  last  expression  becomes  645  X hyp.  log, 

V2  + 41266  .,0,  v , v* +41266  . , , 

■ or  1484  X com.  log . And  here  the 

first  velocity  v being  2000,  the  same  expression  1484  X log. 

— becomes  1484  X log.  of  97-93  = ‘2955  for  the 

height  ascended,  on  this  hypothesis  ; which  was  2920  by  the 
former  problem,  being  nearly  the  same. 

Ex  2.  Supposing  the  same  ball  to  be  projected  with  the 
velocity  of  only  1500  feet.  Then  taking  1 100  velocity,  whose 
tabular  resistance  is  27-6,  being  next  above  the  half  of  that 
for  1500.  Hence,  as  11002  : v2  : : 27  6 : •00002375b2  =avs. 

. . ej)  , av2  *f 

This  value  of  a substituted  in  the  theorem  — X h.  1.  — — - 

4 ga  w 

also  1500  for  v,  and  105  for  w,  it  brings  out  x = 2728  for 
the  height  in  this  case,  being  but  a little  above  the  ratio  of 
the  square  roots  of  the  velocities  2000  and  1500,  as  that  ratio 
would  give  only  2560. 

Ex.  3.  To  find  the  height  ascended  by  the  first  ball  pro- 
jected with  860  feet  velocity.  Here  taking  600,  whose  re- 
sistance 6-69  is  a near  medium  ; then  as  6002  : 6-69  : : 1 : 

•0000186  = a.  Hence  — Xh.l.  — ~ = 2334  the  height  ; 

64a  w 

which  is  less  than  half  the  range  (5100)  at  45°  elevation,  but 
more  than  half  the  range  (4l0(i)  at  15°  elevation,  art.  105  of 
Mot.  and  Forces,  being  indeed  nearly  a medium  between 
the  two. 


Ex. 


OF  GUNNERY. 


461 


Ex.  4.  With  the  same  ball,  and  1640  velocity.  Assume 
1200,  whose  resistance  34  13  is  nearly  a medium.  Then  as 

7/1  n \ 2r  -j-  rjsf 

12002  : 34-13  : : 1 : -0000237  = a.  Hence  c Xh.l.  — -- 

64a  w 

— 2854  ; again  less  than  half  the  range  (6000)  by  experiment 
in  this  vol.  even  with  15°  elevation. 

Ex.  5.  For  any  other  ball  whose  diameter  is  d,  and  its 

weight  ay,  the  resistance  of  the  air  being  =J^"——bd3l- v2 

putting  b = the  retarding  force  will  be 

_ • . ,bd2-v2  4.  w , . — 

thence  — vv  =*  2gx  X , and  x—-g~  X and 

0 w 2 g od2  v 2 -f-w 

_ W w , . bd2v !4w  w , , fe2®s-f» 

the  cor.  flu.  x — - — - X h.  1. =- — - Xh.  1. 

4i{f>d2  bd2v 2 -j-  w i:rbil-  w 

for  the  whole  height  when  v — 0.  Now  if  the  ball  be  a 24 
pounder,  whose  diameter  is  5-6,  and  its  square  31-36  ; then 

= -^-=  1794  : 


bd2  ~ ‘0002091,  and  -- 

oOOOOO  4 gbtl2 


24 


64  bad 


8b  d2 


and  bd2v2  =836,  and 


6.1s  v 2 + w_ 


836  + 24  860  215  , c 

therefore 


24 


24 


215 


x = 1794  X h.  1 — = 1794  X 3 57888  = 6420  ; berng 
6 

more  than  double  the  height  of  that  of  the  small  ball,  or  a little 
more  than  a mile,  and  very  nearly  the  same  as  in  the  2d  ex- 
ample to  prob.  4. 

PROBLEM  VI. 

To  determine  the  Time  of  the  Ball's  ascending  to  the  Height 
determined  in  the  last  prob.  by  the  same  Projectile  Velocity  as 
there  given. 


By  that  prob.  ± = -^X 


-a2  + - 
a 


, . x 7 If 

— .ther.;  = - = - — X 

w v 2 ga  o w 


~ a ’ 

arc  to 


the  fluent  of  which,  by  form  11,  is  — - */  - X 
’ J 2ga  v w 

radius  1 tang.  — ^-=— ./  — X arc  tan.— - ; or  bv  cor- 

w 2g  v a ™ *' 

v~ 


y- 

v n 


rection  t = — «/—  X (arc  tang. — v — arc  tan.  — -Y 
2gv  a x ° _w  to'’ 


the 


a/ 


v/- 


time  in  general  when  the  first  velocity  v is  reduced  to 
v.  And  when  v = 0,  or  the  velocity  ceases,  this  becomes 

for  the  time  of  the  whole 


1 


t — X arc  to  tans 

a 


v/- 


Nc 


.ascent. 


462 


THEORY  AND  PRACTICE 


Now,  as  in  the  last  prob.  v=  2000,  no  — Y05,  a = -000025^ 

= 9of^OO-  HenCe  T = 41266'  and  ✓ J = 203>14>  ai><* 

y 

— — = 98-445  the  tangent,  to  which  corresponds  the  arc 

\/ 
v a 

of  89°  25',  whose  length  is  Y5606  ; then  — X 205-14  X 


1-5606 


203-14X  1'5606 
32 


= 9"-91,  the  whole  time  of  ascent. 


Remark.  The  time  of  freely  ascending  to  the  same  height 
2955  feet,  that  is,  without  the  air’s  resistance,  would  be 

2953 

=1^/2955  = 13''  -59  ; and  the  time  of  freely  as- 
cending, commencing  with  the  same  velocity  2000,  would  be 
v__2000  __  „ _ , „ 

2s-  32  2 2 


PROBLEM  VH. 


To  determine  the  same  as  in  prob.  v,  taking  into  the  ac- 
count the  Decrease  of  Density  in  the  Air  as  the  Ball  ascends  in, 
the  Atmosphere. 

In  the  preceding  problems,  relating  to  the  height  and  time 
of  balls  ascending  in  the  atmosphere,  the  decrease  of  density 
in  the  upper  parts  of  it  has  been  neglected,  the  whole  height 
ascending  by  the  ball  being  supposed  in  air  of  the  same  den- 
sity as  at  the  earth’s  surface.  But  it  is  well  known  that  the 
atmosphere  must  and  does  decrease  in  density  upwards,  in  a 
very  rapid  degree  ; so  much  so  indeed,  as  to  decrease  in  geo- 
metrical progression  : at  altitudes  which  rise  only  in  arithme- 
tical progression  : by  which  it  happens,  that  the  altitudes 
ascended  are  proportional  only  to  the  logarithms  of  the  de- 
crease of  density  there.  Hence  it  results,  that  the  balls  must 
be  always  less  and  less  resisted  in  their  ascent,  with  the  same 
velocity,  and  that  they  must  consequently  rise  to  greater 
heights  before  they  stop.  It  is  now  therefore  to  be  consi- 
dered what  may  be  the  difference  resulting  from  this  circum- 
stance. 

Now,  the  nature  and  measure  of  this  decreasing  density, 
of  ascents  in  the  atmosphere,  has  been  explained  and  deter- 
mined in  prop.  76,  Pnuematics.  It  is  there  shown,  that 
if  d denote  the  air’s  density  at  the  earth’s  surface,  and 
d its  density  at  any  altitude  a,  or  a;  then  is  x = 63551  X 

log.  of  ~ in  feet,  when  the  temperature  of  the  air  is  55°  ; 

and  60000  X log.  D,  for  the  temperature  of  freezing  cold ; 

we 


OF  GUNNERY. 


463 


we  may  therefore  assume  for  the  medium  x — 62000  X log.- 

for  a mean  degree  between  the  two. 

But  to  get  an  expression  for  the  density  d , in  terms  of  x 
out  of  logarithms,  without  woich  it  could  not  be  introduced 
into  the  measure  of  the  ball’s  resistance,  in  a manageable  form 
we  find  in  the  first  place,  by  a neat  approximate  expression 

for  the  natural  number  to  the  log.  of  a ratio  whose  terms 

do  not  greatly  differ,  invented  by  Dr.  Halley,  and  explained 

in  the  Introduction  to  our  Logarithms,  p.  110,  that  . ,,  X » 

ni-il 

nearly,  is  the  number  answering  to  the  log.  I of  the  ratio^, 
where  n denotes  the  modulus  -43429448  &c.  of  the  common 
logarithms.  But,  we  before  found  that  x—  62000  X log.  of  ~> 


DC  D 

or  is  the  log.  of  which  log.  was  denoted  by  l in  the 

62000  ° d * J 

expression  just  above,  for  the  number  whose  log.  is  l or 


62000 


; substituting  therefore 


62000 


for  l,  in  the  expression 


124000 


X n = d or. 


u 

-t-~  X d,  it  gives  the  natural  number 
n+\l  70  I ~ 

nA — 

124000 

124000a — x __  , jjje  (jengjty  0f  the  air  at  the  altitude  x,  put- 
124000n-f-  x > J ’f 

ting  d = 1 the  density  at  the  surface.  Now  put  124000a  or 

nearly  54000  = c ; then  C~~  will  be  the  density  of  the  air 

at  any  general  height  x 

But,  in  the  5th  prob.  it  appears  that  av-  denotes  the  re- 
sistance to  the  velocity  v,  or  at  the  height  x,  for  the  density 
of  air  the  same  as  at  the  surface,  which  is  too  great  in  the 

ratio  of  c 4-  x to  c — x ; therefore  av2  X - — - will  be  the 

c-f  x 

resistance  at  the  height  x,  to  the  velocity  v,  where  a 
•000025^.  To  this  adding  w,  the  weight  of  the  ball,  gives 

..  . c — x 


C “j-  ^ 


4-  w for  the  whole  resistance,  both  from  the  air 


d"ij2  c—X  TV 

and  the  ball’s  mass  ; conseq.  — X -f-  —will  denote 

w C+X  %!) 

the  accelerating  force  of  the  ball.  Or,  if  we  include  the 

small  part  ™ or  1,  within  the  factor  — , which  will  make 
■w  c-\-x 

no  sensible  difference  in  the  result,  but  be  a great  dehl  simpler 


464 


THEORY  AND  PRACTICE 


V,  ,,  ■ av2  +w  w c — x . . 

in  the  process,  then  is  — — — X = f the  accelerating 

force.  Conseq.  — v'v  = 2 gfi  = 2 gx  X C~X  x ani{ 

C -f-  x 10 

I C — x . W . — Vv  , 1-  • • . , 2c 

hence  — — x — — — -r- — > or  by  division,  — x H 

c+x  %g  av2  + w J ’ ' c -f  x 

t — “ X ~ri 
32a  , w 

V2  + — 


Now  the  fluent  of  the  first  side  of  this  equation  is  evi- 
dently — a-  + 2 c X h.l.  (c+.r)  ; and  the  fluent  of  the  latter 

. 'll)  cjju 

side,  the  same  as  in  prob.  5,  is  — — X h.  1.  (v-  + there- 
fore  the  general  fluential  equa.  is  — x + 2c  X h.  1.  (c+x)  = 

— — X h 1.  (v2  +— ).  But  when  a = 0,  and  v = v the  initial 

64a  v a 

velocity,  this  becomes  0+2cXh.  1.  c =(.~  X h.  1 (v2  -f-  — ) ; 

theref.  by  subtraction  the  correct  fluents  are  - x -f  2c  X h.  I. 

X h.  1.  ar-  ^ W,  when  the  first  velocity  v is  dimi- 
c 64a  av2  *r  w 

nished  to  any  less  one  v ; and  when  it  is  quite  extinct,  the 
state  of  the  fluents  becomes  — x + 


2c  Xh.1.^1^  X 
c 64a 


h.  1.  — — for  the  greatest  height  x ascended. 

C "f-  cc 

Here,  in  the  quantity  h.  1.  — - — , the  term  x is  always  small 
in  respect  of  the  other  term  c ; therefore,  by  the  nature  of 


logarithms,  the  h.  1.  of  —^is  nearly 


x 2x  , . 

-rrOf,,  ; theref. 
2c+x 


the  above  fluents  become 

= ixu  “vJ+'” 


, 4cx  2cx_  x2 

X ~2c+x  2c-f-x 


2 c — x 


2c  q.  x 

Now  the  latter  side  of  this  equation  i^ 

64a  w 

the  same  value  for  x as  was  found  in  the  5th  problem,  wlficli 
therefore  put  = b ; then  the  value  of  x will  be  easily  found 

2C  ^ cc 

from  the  formula  -- , x = b,  by  a quadratic  eouation.  Or, 
2c-px 

still  easier,  and  sufficiently  near  the  truth,  by  substituting  b 

• 2c— x 

for  x in  the  numerator  and  the  denominator  of  — — then 

2 c+x 

2c — b . . , 2c-f  b , , . 

x — b,  and  hence  x = b,  or  by  proportion,  as 

2 c+b  2c — b Jftr 

2c  — b : 2c  + b : : b : x ; that  is,  only  increase  the  value  of 
x,  found  by  prob.  5,  in  the  ratio  of  2c  — b to  2 c+6 

Now,  in  the  first  example  to  that  prob.  the  value  of  x or 

b was 


OF  GUNNERY. 


465 


b was 'there  found  = 2955  ; and  2c  being  = 108000,  tberef. 
2c — 6 = 105045,  and  2c  + b — 110955,  then  as  105045  : 
110955  : : 2955  : 3121  = the  value  of  the  height  x in  this 
case,  being  only  166  feet  or,  T’jth  part  more  than  before. 

Also,  for  the  2d  example  to  the  5th  prob.  where  x was  = 
6420  ; therefore  as  2c—  b : 2c  + b or  as  105045  : 119055 
6420  : 6780  the  height  ascended  in  this  example,  being  also 
the  18th  part  more  than  before.  And  so  on,  for  any  other  ex- 
amples ; the  value  of  2c  being  the  constant  number  108000. 

PROBLEM  VIII. 


To  determine  the  Time  of  a Ball's  Ascending,  considering  the 
* Decreasing  Density  of  the  Air  as  in  the  last  prob. 


The  fluxion  of  the  time  is  i = i.  But  the  general  equa- 

V 

tion  of  the  fluxions  of  the  space  x and  velocity  v,  in  the  last 

' ; ther.  i =”+r^X 

32  c—x  or2 -f<ia 

But  x,  which  is  al- 


prob.  was  — r-i  = — X 
r c-j-x  32  av  2-fw 

, . x iv  c-Lx 

hence  t = - = — X — X . 

v 32  c — x avs  -f-iu 

ways  small  in  respect  of  c,  is  nearly  = b as  determined  in  the 

C c ~j  3? 

iast  problem  ; theref.  — - may  be  substituted  for  — - with- 


out sensible  error  ; and  then  } becomes  X X — . 

32  c-b  av2+ w 

Now,  this  fluxion  being  to  that  in  prob.  6,  in  the  constant  ratio 
of  c—b  to  c b,  their  fluents  will  be  also  in  the  same  con- 
stant ratio  But,  by  the  last  prob.  c — 54000,  and  b — 2955 
for  the  first  example  in  prob.  5 ; therefore  c—b~  51045,  and 
s + b = 56955,  also,  the  time  in  problem  6 was  9"*9 1 ; there- 
ore  as  51045  : 56955  ::  9"- 91  : 1 1"'04  for  the  time  in  this  case 
oeing  l"1 13  more  than  the  former,  or  nearly  the  9th  part  more  ; 
vhich  is  nearly  the  double,  or  as  the  square  of  the  difference, 
n the  last  prob.  in  the  height  ascended. 


PROBLEM  IX. 


To  determine  the  circumstances  of  Space,  Time,  and  Veloci- 
y,  of  a Ball  Descending  through  the  Atmosphere  by  its  own 
'Veight. 

It  is  here  meant  that  the  balls  are  at  least  as  heavy  as  cast 
roD,  and  therefore  their  loss  of  weight  in  the  air  insensible  ; 
md  that  their  motion  commences  by  their  own  gravity  from  a 
tate  of  rest.  The  first  object  of  enquiry  may  be,  the  utmost 
legree  of  velocity  any  such  ball  acquires  by  thus  descending, 
'fow  it  is  manifest  that  the  ball’s  motion  is  commenced,  and 
tniformly  increased,  by  its  own  weight,  which  is  its  constant 
irging  force,  being  always  the  same,  and  producing  an  equal 
Vol.  II.  ' 60  increase 


466 


THEORY  AND  PRACTICE 


increase  of  velocity  in  equal  times,  excepting  for  the  diming 
tion  of  motion  by  the  air’s  resistance.  It  is  also  evident  that 
this  resistance,  beginning  from  nothing,  continually  increases, 
in  some  ratio,  with  the  increasing  velocity  of  the  ball  Now, 
as  the  urging  force  is  constantly  the  same,  and  the  resisting 
force  always  increasing,  it  must  happen  that  the  latter  will  at 
length  become  equal  to  the  former*  : when  this  happens, 
there  can  afterwards  be  no  farther  acceleration  of  the  motion, 
the  impelling  force  and  the  resistance  being  equal  and  the  ball 
must  ever  after  descend  with  a uniform  motion.  It  fdlows 
therefore  that,  to  answer  the  first  enquiry,  we  have  only  to  de- 
termine when  or  what  velocity  of  the  ball  will  cause  a resist- 
ance just  equal  to  its  owd  weight. 

Now,  by  inspecting  the  tables  of  resistances  preceding  prob. 
1,  particularly  the  first  of  the  three  tallies,  the  weight  of  the 
ball  being  105  lb.  we  perceive  that  the  resistance  increases 
in  the  2d  column,  till  0”69  opposite  to  200  velocity,  and  1 56 
answering  to  300  velocity,  between  which  two  the  proposed 
resistance  105,  and  the  correspondent  velocity,  fall.  But,  in 
two  velocities  not  greatly  different,  the  resistances  are  very 
nearly  proportional  to  the  squares  of  the  velocities  There- 
fore, having  given  the  velocity  200  answering  to  the  resistance 

0- 69,  to  find  the  velocity  answering  to  the  resistance  1 05,  we 

must  say,  as  0-69  : 1 05  : : 2002  : v3  - 60870,  theref  v = 

x / 60870  = 246,  is  the  greatest  velocity  this  ball  can  acquire  ; 
after  which  it  will  descend  with  that  velocity  uniformly,  or 
at  least  with  a velocity  nearly  approaching  to  246. 

The  same  greatest  or  uniform  velocity  will  also  be  directly 
found  from  the  rule  *1)0001 725v2  = r,  near  the  end  of  pro- 
blem 2,  where  r is  the  resistance  to  the  velocity  v,  by  making 

1- 05  — r \ for  then  v3  — —— = 60870,  the  same  value 

’ -0000172o 

for  v2  as  before. 

But  now  , for  any  other  weight  of  ball  ; as  the  weights  of 
the  balls  increase  as  the  cubes  of  their  diameters,  and  their 
resistances,  being  as  the  surfaces,  increase  only  as  the  squares 
of  the  same,  which,  is  one  power  less  ; and  the  resistances 
being  also  in  this  case,  as  the  squares  of  the  velocities,  we 
must  therefore  increase  the  squares  of  the  velocity  in  the 
ratio  of  the  diameters  of  the  balls  ; that  is,  as  1*965  : d : : 

2462  : — 6 d — v2  and  hence  v = 246  */  = 1754  ./  d. 

1*965  1*965  ‘ v 

If  we  take  here  the  31b  ball,  belonging  to  the  2d  table  of 
resistances,  whose  diameter  d is  — 2*80  ; then^/2*80=l*673} 
and  1754  X 1 67  = 294,  is  the  greatest  or  uniform  velocity 
with  which  the  31b  ball  will  descend.  And  if  we  take  the 

* This  reasoning  is  not  conclusive-  The  velocity  of  the  descending 
body  increases  continually,  but  never  becomes  equal  to  a certain  de- 
terminate velocity- 


OF  GUNNERY. 


4&1 

felb  ball,  whose  diameter  is  3-53  inches,  as  in  the  3d  table  of 
resistances  : then- y'  3*53  = 1-88,  and  175|  X 1S8  = 330, 
being  the  greatest  velocity  ttiat  can  be  acquired  by  the  dlb 
ball,  and  with  which  it  will  afterwards  uniformly  descend. 
For  a dlb  ball,  whose  diameter  is  4 04,  the  velocity  will  be 
1751  x 2 01  — 353.  And  so  on  for  any  other  size  of  iron 
ball,  as  in  the  following  table.  Where  the  first  column  con- 
tains the  weight  of  the  balls 
in  lbs  ; the  2d  their  diame- 
ters in  inches  ; the  3d  their 
velocities  to  which  they 
li  nearly  approach,  as  a limit, 
and  therefore  called  their 
terminal  or  last  velocities, 
with  which  they  afterward 
' descend  uniformly  ; and  the 
; 4th  or  last  column  the 
! heights  due  to  these  veloci- 
' ties,  or  the  heights  from 
| which  the  balls  roust  descend 
in  vacuo  to  acquire  them. 

But  it  is  manifest  that  the 
balls  can  never  attain  exactly 
to  these  velocities  in  any 
finite  time  or  descent,  being 
only  the  limits  to  which  they  continually  approach,  without 
ever  really  reaching,  though  they  arrive  very  nearly  at  them 
in  a short  space  of  time  ; as  will  appear  by  the  following  cal- 
culation. 

To  obtain  general  expressions  for  the  space  descended,  and 
: the  time  of  the  descent,  in  terms  of  the  velocity  v : put  x — 
any  space  descended,  t ==  its  time,  and  v the  velocity  ac- 
quired, the  weight  of  the  ball  w = 1-05  lb.  Now,  by  the 
theorem  near  the  end  of  prob.  2,  which  is  the  proper  rule  for 
. this  case,  the  velocity  being  small,  00001  725zj2  = cv2  is  the 
resistance  due  to  the  velocity  v ; theref.  w—cv2  is  the  impelling 

force,  and  - — f the  accelerating  force  ; conseq.  or 

2gfx  = 2gx  X — — - — . and  i = — X — - — , the  correct  flu- 
ent  of  which,  by  the  8th  form,  is  x — ~ X h.  1.  — — — 

<70—  CV- 

the  general  value  of  the  space  x descended. 

Here  it  appears  that  the  denominator  w — cv2  decreases  as 
v increases  ; conseq.  the  whole  value  of  r,  the  descent,  in- 
creases with  v,  till  it  becomes  infinite,  when  the  resistance 

cvs 


VVt. 

lbs. 

Diam. 

inch 

Term. 

Veloc. 

feet. 

* 

Height 
due  to 
v,  feet. 

1 

1-94 

244 

930 

2 

2-45 

275 

1182 

3’ 

2 80 

294 

1260 

4 

3-08 

308 

1432 

6 

3’53 

330 

1701 

9 

404 

353 

1958 

12 

4-45 

370 

2139 

18 

5-09 

396 

2450 

24 

5-60 

415 

2691 

32 

6-17 

436 

2790 

36 

6'41 

444 

3080 

42  1 

6-75 

456 

r>r.< 

468 


THEORY  AND  PRACTICE 


cv2  is  — w the  weight  of  the  ball,  when  the  motion  become § 
uniform  as  before  remarked.  We  may  however  easily  assign 
the  value  of  x a little  before  the  velocity  becomes  qniform, 
or  before  cv 2 becomes  = w.  Thus,  when  cv2  = w,  then 
v — 246,  as  found  in  the  beginning  of  this  problem.  Assume 
therefore  v a little  less  than  that  greatest  velocity,  as  for  in- 
stance 240  : then  this  value  of  v substituted  in  the  general 
formula  for  x above  deduced,  gives  x = 2781  feet,  a little 
before  the  motion  becomes  uniform,  or  when  the  velocity  has 
arrived  at  240,  its  maximum  being  246. 

In  like  manner  is  the  space  to  be  computed  that  will  be 
due  to  any  other  velocity  less  than  the  greatest  or  terminal 
velocity  On  the  contrary,  to  find  the  velocity  due  to  any 

proposed  space  x,  from  the  formula  x — — X h.  1 . — W — 
r 1 r 4 gc  -w — cv  - 

Here  x is  given,  to  find  v.  First  then  — -x=h.  1. — — — • 

iv  w — CV2 

take  therefore  the  number  to  the  hyp.  log.  of  — which 

IV 

number  call  n : then  n = ; conseq.  ktb  — ncu®  = ' 

’ — Cv 1 

and  nw  — 'W  = n cv2 , and  v = ^/  — — w,  a general  theorem 

for  the  value  of  v due  to  any  distance  x.  Suppose,  for  in- 
stance, x is  1000.  Now  4g  being  = 64,  w = 105,  and 

^aCX 

c — -00001725;  theref.  - — = 1 0514,  and  the  natural 

w 

number  belonging  to  this,  considered  as  an  hyp.  log.  i3 
2 8617  = n;  hence  then  v = */  - — w = 199,  is  the  velo- 

V NC 

city  due  to  the  space  1000,  or  when  the  ball  has  descended 
1000  feet. 

Again,  for  the  time  t of  descent  : here  / — but 

V 

x = ~ X — — , as  found  above,  theref.  t — — — 

2g  iv  — cv  2 2g  w—cva 


1 y/-+T> 

the  fluent  of  which  is  — — X h.  1 . — 2 , the  general 

c ,iv 

a/ v 

c 

value  of  the  time  t for  any  value  of  the  Telocity  v ; which 

py 

value  of  t evidently  increases  as  the  denominator  ~ — v 

decreases,  or  as  the  velocity  v increases  ; and  consequently 
the  time  is  infinite  when  that  denominator  vanishes,  which 

is 


OF  GUNNERY. 


469 


is  when  v = 1/  7’  or  cv2  = w,  the  resistance  equal  to  the 

ball’s  weight,  being  the  same  case  as  when  the  space  x be- 
comes infinite,  as  above  remarked.  But,  like  as  was  done 
for  the  distance  x as  above,  we  may  here  also  find  the  value 
of  t corresponding  to  any  value  of  v,  less  than  its  maximum 
246,  and  consequently  to  any  value  of  x,  as  when  v is  240  for 
instance,  or  x — 2781,  as  determined  above.  Now,  by  sub- 
stituting 240  for  v,  in  the  general  formula. 


4g 


v/  7 X 


h.  1.— 


.w  i 

\/t  + w 


v'- 

v r 


-,  it  brings  out  t = 16"-575  ; so 


that  it  would  be  nearly  16A  seconds  when  the  velocity  arrives 
at  240,  or  a little  less  than  tne  maximum  or  uniform  degree, 
viz.  246,  or  when  the  space  descended  is  2781  feet. 

Also,  to  determine  the  time  corresponding  to  the  same,  or 
when  the  descent  is  1000  feet,  or  the  velocity  199  : find  the 
, c 1 , w _ 1 , Hg  _ 246  _ 123 

value  ot  Aa_  v'  ^ fi4  v -00001725  64  32 


4 g 


Then 


64  v -00001725 
^ c = ■ the  hyp.  log.  of  which  is  2-2479. 

^7~ 

Hence  2-2479  X 


246—159  47 


123 

71 


= 8"-64,  the  time  of  descending  1000 

feet,  or  when  the  velocity  is  199. 

See  other  speculations  on  this  problem,  in  Probi  22,  Pro- 
jectiles, as  determined  from  theory,  viz.  without  using  the 
experimented  resistance  of  the  air. 

PROBLEM  X, 

To  determine  the  Circumstances  of  the  Motion  of  a Ball  pro - 
jected  Horizontally  in  the  Air;  abstracted  from  its  Vertical 
Descent  by  its  Gravitation. 

Putting  d for  the  diameter,  and  w the  weight  of  the  ball, 
v the  velocity  of  projection,  and  v the  velocity  of  the  ball 
after  having  moved  through  the  space  x Then  by  corol.  i 
to  prob.  2,  if  the  velocity  is  considerable,  such  as  usual  ia 
practice,  the  resistance  of  the  ball  moving  with  the  velocity 

UlH)  — 71T)  . • 

v , is  (mv2  — nv)  d2 , and  therefore d 2 is  the  retardive 

7 \ ' w 

force /;  hence  the  common  formula  vv  — 2 gff,  is  — vv  = 


32i  X 


?73D-  — nv 


w 

w 


- d2 , and  theref.  x = 


32  d* 


mv-1  — nv 


32  d? 


X 


;mi — n 32d2.ni 


X 


the  fluent  of  which  is  obviously 


74) 


470 


THEORY  ANE>  PRACTICE 


- — , X — hyp.  log.  of  v , and  by  the  correction  by  the 

32md2  Jf  b m J J 


first  velocity  v,  it  becomes  x 


10 


32inU2 


X h.  log. 


, the 


general  formula  for  the  distance  passed  ever  in  terms  of  the 
velocity 

Now.  for  an  application,  let  it  be  required  first,  to  deter1 
mine  in  what  space  a 24lb  ball  will  have  its  velocity  reduced 
from  1780  feet  to  1500,  that  is  losing  280  feet  of  its  first 
velocity.  Here,  d = 5 6,  w = 24,  v = 1780,  and  v = 1500  ; 

also  — = 150.  Hence  J—r  = 3587  4,  then  x = 3587  4 X 

m 16  md2 

h.  1.  = 3587-4  X h.  1.  1^-  =3587-4  Xh.  l.£g  = 

v — 150  13o 

676  feet,  the  space  passed  over  when  the  ball  has  lost  280  feet 
of  its  motion. 

Again  to  find  with  what  velocity  the  same  ball  will  move, 
after  having  described  1000  feet  in  its  flight.  The  above, 

theorem  is  x or  1000  = 3587  4 X h.  1.  - — ^ = 3587-4  X 


h.  1. 


— 150 

1630  10000  1630  . ..  . 4 

, or  „co„A  = h.  1.  - — — ; but  the  number  to  the 


<b—  150’  " 35874 
10000  . 


7)  — 150 


1630 


hyp.  log.  -^g y4  is  1-7416  = n suppose  ; then  — — , and 
sv  — 150n  = 1630,  or  av  ~ 1630  -f*  150n,  and  v = — — — 

N 

350  = 936  150  = 786,  the  velocity,  when  the  ball  has 

moved  1000  feet. 

Next,  to  find  a theor.  for  the  time  of  describing  any  space, 
&r  destroying  any  velocity  : Here  l = ~ = X — - — r— 


the  fluent  of  which,  by  the  9th  form  is  t = 

V IV 


X 


k.  1. 


32nd2 


X h.  1. 


32md2 

, and  by  correction 


t = 


v - 

1U 

32nd2 


X (h.  1.- 


h.  1.- 


m 

v 


)=t^t ; xh*P-  IoS- 


32nd2 


' ~ putting  v for  the  first  velocity,  and  150  for  — 

its  value,  as  before. 

Now,  to  take  for  an  example  the  same  24lb  ball,  and  its 

projected 


OF  GUNNERY. 


472 


projected  velocity  1780,  as  before  ; let  it  be  required  to  find 
in  what  time  this  velocity  will  be  reduced  to  786.-  Here  then 
v = 1780,  v = 786, tv  = 24 ,d  = 5-6,  d2  = Si -36,  n=-001  ; 

hence  ■ „■  — 


32nd 

1353 


750 

31-36 


= 23-916  ; and 


v — 150 
v — 150  v 


- = 1^2  x 


636 


786 

1780  18868’ 

•1099  = 2"  628,  the  time  required. 


— , the  hyp.  log.  of  which  is  -1099  ; then  31-36  X 


For  another  example,  let  it  be  required  to  find  when  the 
velocity  will  be  reduced  to  1000,  or  780  destroyed.  Here 
v = 1000,  and  all  the  other  quantities  as  before.  Then 

v — 150  w v 1630  , 1000'  1630  , . . , . , . 

X - = — — X — — = , the  hyp.  log.  of  which  is 

— 150  v 850  1 80  1513  J ^ ° 

•07449  ; theref.  3T36  X 07449  = 1'  -78,  is  the  time  sought. 

On  the  other  hand,  if  it  be  required  to  find  what  will  be 
the  velocity  after  the  ball  has  been  in  motion  during  any  given 
time,  as  suppose  2 seconds,  we  must  reverse  the  calculation 

thus  : t — 2"  being  = X h.  1.  — . -=  23-916  X 


32nd2 


v— 150  v 


h.  1. 


v — 150 
d— 150 


— : theref.  — -- — = -083626  is  the  hyp.  log  of 
v ’ 23-yi6  ° 


v- — ^ — , the  number  answering  to  which  is  T08725 
v— 150  v ° 


v— 150 
^150  ‘ 
150nv 


— . Hence  nvv  — 150 


290290  ncl  . 

: — 951,  the  velo- 

o05'o03 


suppose,  that  is,  w 
vv—  1 50x>,  and  v — 

15 0 -J-  nv  — v 
city  at  the  end  of  2 seconds. 

The  foregoing  calculations  serve  only  for  the  higher  velo- 
cities, such  as  exceed  200  or  300  feet  per  second  of  time. 
But,  for  those  that  are  below  300,  the  rule  is  simpler,  as  the 
resistance  is  then,  by  cor  2 prob.  2,  •00000447iZ:ii>2  = cd2v2 , 
where  d denotes  the  diameter  of  any  ball.  Hence  then, 

employing  the  same  notation  as  before,  =f,  and  — v-u  = 


32 fx  — 32j?  X 


aU  v2 


theref.  x — 


•w  ..  — v 


3 2cd3 


X 


-,  the  correct 


X h. 


1.1. 


fluent  of  which  is  x — - 

32 cd2  v 

Now,  for  an  example,  suppose  the  first  velocity  to  be 
300  = v,  and  the  last  v = 100  for  a 241b  ball.  Then 
w = 24,  d = 5 6,  d2  — 31-36,  c = -00000447  ; therefore 

= — = 5350  ; and  — = = 3,  the  hyp  log, 

32cd2  125-44 c ’ v 100  ’ 2 V a 

of  which  is  1-0986  ; theref.  1 0986  X 5350  — 5878  = x,  is 
the  distance. — If  the  first  velocity  be  only  200  = v j then 


472 


THEORY  AND  PRACTICE 


- =;  2,  the  hyp.  log.  of  which  is  '69315,  therefore.  69315  X 

V 

5350  = 3708  = x,  the  distance. 

And  conversely,  to  find  what  velocity  will  remain  after 
passing  over  any  space,  as  4000  feet  the  first  velocity  being 

ti  , * , ~ v . x 4000  400 

v = 200.  Here  the  hyp.  log.  ol  — is  — = — — = — 

h v 5350  5350  535 

80 

— — = -74766,  the  natural  number  of  which  is  2-1120, 
io  7 

v v £00 

that  is,  2-112  = — ; therefore  v — — ■ = — - = 947,  the 
velocity. 


Again,  for  the  time  t : since  x 


~Ti  X , therefore 

yicd2  tj 


} = — = X — , the  correct  fluent  of  which  is 

v 32  ccP  ti’ 

nv  . . , 1 1 \ nu  v — v „ . 

t = — — X ( ) =- X • — So,  lor  example 

32C£/2  V-  " ' r 


32ca2 


if  v 


200 


300,  and  v = 100  : then  — * = ~~~  = ; then 

’ ’ w 30000  300 

or  5350  X — = 35''  f — t,  the  time  of  reducing  the 
32c</2  300  3 

300  velocity  to  100,  or  of  passing  over  the  space  5878  feet. 
And,  reversing,  to  find  the  velocity  v,  answering  to  any 

given  time  t : Since  t = — — X f- -)  = 5350  x 

32ca3  v v v ' 

5jJ°  Here,  if  t be  given  .=  30 


(— — ) theref.  v — 

v 5 oSO-f-lv 


and  v = 300  : then  v 


1 12,  the  velocity  sought. 


5350tj  535  w 32100 

1 * 1 • ■ --  ■ x oUU  ■■■  - — 

5350+9000  1435  287 


Corol.  The  same  form  of  theorem,  x = ■——  X h.  1.— 

32crf2  v 

as  above,  is  brought  out  for  small  velocities,  will  also  serve 
for  the  higher  ones,  if  we  employ  the  medium  resistance  be- 
tween the  two  proposed  velocities,  as  was  done  in  prob  6. 
Thus,  as  in  the  first  example  of  this  problem,  where  the  two 
velocities  are  1780  and  1500,  the  resistance  due  to  the  velo- 
city 1700,  in  the  first  table  of  resistances,  being  74-13,  say  as 
17002  : 1780s  : : 74'13  : 8T27,  the  resistance  due  to  the  ve- 
locity 1780  ; then  the  mean  between  8T27  and  57  25,  due 
to  1500  velocity,  is  69-26.  or  rather  take  69+  Again,  as 
y/  65-7  : y/  69i  : : 1600  : 1646,  the  velocity  due  to  the.  me- 
dium resistance  69+  Hence,  as  in  prob.  5,  as  16462  : v2  : : 
69£  -.  •00002565,a2  = suppose  av2,  the  resistance  due  to  any 

velocity 


©F  GUNNERY. 


473 


velocity  v,  between  1780  and  1500,  for  the  1 *05lb  ball.  And 
as  1-9652  : 5 62  : : av2  : 8-124an2  — -00020838t>2  — bv2  sup- 
pose, the  resistance  due  to  the  same  velocity,  with  the  2 lib 

ball.  Therefore  ^ =*=/,  and  — vj,  — 32 fx  — f bv2x,  and 


To  determine  by  theory,  the  trajectory  a projectile  de- 
scribes in  the  air,  is  one  of  the  most  difficult  problems  in  the 
whole  course  of  dynamics,  even  when  assisted  by  all  the  ex- 
periments that  have  hitherto  been  made  on  this  branch  of 
physics  ; and  is  indeed  much  too  difficult  for  this  place  in 
: the  full  extent  of  the  problem  : the  consideration  of  it  must 
therefore  be  reserved  for  another  occasion  when  the  nature 
of  the  air’s  resistance  can  be  more  amply  discussed.  Even 
the  solutions  of  Newton,  of  Bernoulli,  of  Euler,  of  Borda 
&c.  &c.  after  the  most  elaborate  investigations,  assisted  by  all 
the  resources  of  the  modern  analysis,  amount  to  no  more 
than  distant  approximations,  that  are  rendered  nearly  useless 
even  to  the  speculative  philosopher,  from  the  assumption  of 
a very  erroneous  law  of  resistance  in  the  air,  and  much  more 
so  to  the  practical  artillerist,  both  on  that  account,  and  from 
the  very  intricate  process  of  calculation,  which  is  quite  inap- 
plicable to  actual  service.  The  solution  of  this  problem  re- 
quires, as  an  indispensible  datum,  the  perfect  determination 
by  experiment  of  the  nature  and  laws  of  the  air’s  resistance 
at  different  altitudes,  to  balls  of  different  sizes  and  densities 
moving  with  all  the  usual  degrees  of  celerity.  Unfortunately 
however,  hardly  any  experiments  of  this  kind  have  been 
made  excepting  those  which  on  some  occasions  have  been 
published  by  myself,  as  in  my  tracts  of  1786,  as  well  as  in 
my  Dictionary,  some  few  of  which  are  also  given  in  art  105  of 
Mot.  and  Forces,  with  some  practical  inferences.  And 
though  I have  many  more  yet  to  publish,  of  the  same  kind, 
much  more  extensive  and  varied,  I cannot  yet  undertake 
to  pronounce  that  they  are  fully  adequate  to  the  purpose  in 
hand. 

All  that  can  be  here  done  then,  in  the  solution  of  the  pre 
sent  problem,  besides  what  is  delivered  in  this  volume,  is 
to  collect  together  sense  of  the  best  practical  rules,  founded 


3 v 3 

the  correct  fluent  of  which  is  -X  h.  1.  — 

46  v 46 


X h.  1.  ??  = 3600  X -171148  = 616  the 


velocity  sought. 


PROBLEM  XI. 


To  determine  the  Ranges  of  Projectiles  in  the  Air. 


Vol.  II. 


61 


partly 


474 


THEORY  AND  PRACTICE 


partly  on  theory,  and  partly  on  practice.  1.  In  the  first  place 
then,  it  may  be  remarked,  that  the  initial  or  first  velocity  of 
a ball  may  be  directly  computed  by  prob.  17,  page  393  of 
this  -volume  ; having  given  the  dimensions  of  the  piece,  the 
weight  of  the  ball,  and  the  charge  of  powder.  Or  otherwise, 
the  same  may  be  made  out  from  the  table  of  experimented 
ranges  and  velocities  in  pa.  141  of  this  volume,  by  this  rule, 
that  the  velocities  to  different  balls,  and  different  charges  of 
powder,  are  as  the  square  roots  of  the  weights  ef  the  powder 
directly,  and  as  the  square  roots  of  the  weights  of  the  balls  in- 
versely. Thus,  if  it  be  enquired,  with  what  velocity  a 24lb 
ball  will  be  discharged  by  81b  of  powder.  Now  it  appears  in 
the  table,  that  8 ounces  of  powder  discharge  the  lib  ball  with 
1640  feet  velocity  ; and  because  81b  are  — 128  ounces  ; 
therefore  by  the  rule,  as  | : y/  ^ : : 1640  : 1640.^/if 
— 1640^1-  = 1339,  the  velocity  sought.  Or  otherwise, 
by  rule  1,  p.  142  of  this  vol.  as  24  : 16  : : 1600  : 1306, 

the  same  velocity  nearly.  But  when  the  charges  bear  the 
same  ratio  to  one  another  as  the  weight  of  the  balls,  that  is 
when  the  pieces  are  said  to  be  alike  charged,  then  the  veloci- 
ties will  be  equal.  Thus,  the  lib  ball  by  the  2 oz  charge  be- 
ing the  8th  part  of  the  weight,  and  the  24lb  ball,  with  31b  of 
powder,  its  8th  part  also,  will  have  the  same  velocity,  viz. 
860  feet.  In  like  manner,  the  1230  tabular  velocity,  answer- 
ing to  4 oz  of  powder,  the  4th  part  of  the  ball,  will  equally 
belong  to  the  24lb.  ball  with  61b  of  powder,  being  its  4th  part 
and  the  tabular  velocity  1640,  answering  to  the  8oz  charge, 
which  is  i the  weight  of  ball,  will  equally  belong  to  the  24lb 
ball  with  121b  of  powder,  being  also  the  4-  of  its  weight 

2.  By  prob.  9 will  be  found  what  is  called  the  terminal  ve-  • 
locity,  that  is,  the  greatest  velocity  a ball  can  acquire  by  des-  i 
cending  in  the  air  ; indeed  a table  is  there  given  of  the  seve- 
ral terminal  velocities  belonging  to  the  different  balls,  with  the 
heights,  in  an  annexed  column,  due  to  those  velocities  in  vacuo, 
that  is  the  heights  from  which  a body  must  fall  in  vacuo,  to 
acquire  those  velocities. 

3.  Given  the  initial  velocity,  to  find  the  elevation  of  the 
piece  to  have  the  greatest  range,  and  the  extent  of  that  range. 
These  will  be  found  by  means  of  the  annexed  table,  altered 


from 


OF  GUNNERY. 


475 


from  Professor  Robison’s 
in  the  Encyclopaedia  Bri- 
tannica,  and  founded  on  an 
approximation  of  Sir  I. 

Newton’s.  The  numbers 
in  the  first  column,  multi- 
plied by  the  terminal  velo- 
city of  the  ball,  give  the 
initial  velocity  ; and  the 
numbers  in  the  last  co- 
lumn, being  multiplied  by 
the  height,  give  the  great- 
est ranges  ; the  middle  co- 
lumnshowing the  elevations 
to  produce  those  ranges. 

To  use  this  table  then, 
divide  the  given  initial  ve- 
locity by  the  terminal  ve- 
locity peculiar  to  the  ball, 
found  in  the  table  in  prob. 

9,  and  look  for  the  quo- 
tient in  the  first  column 
here  annexed.  Against 
this,  in  the  2d  column  will 
be  found  the  elevation  to 
give  the  greatest  range  ; and  the  number  in  the  3d  column 
multiplied  by  a,  the  altitude  due  to  the  terminal  velocity, 
also  found  in  the  table  in  problem  -9,  will  give  the  range, 
nearly. 

Ex.  1.  Let  it  be  required  to  find  the  greatest  range  of  a 
24lb  ball,  when  discharged  with  1640  feet  velocity,  and  the 
corresponding  angle  to  produce  that  range.  By  the  table  in 
prob.  9,  the  terminal  velocity  of  the  24lb  ball  is  415,  and  its 

1640 

producing  altitude  2691  : hence  ^ — = 3 95,  nearly  equal  to 

3-9865  in  the  1st  column  of  our  table,  to  which  corresponds 
the  angle  34°  15’,  being  the  elevation  to  produce  the  greatest 
range  ; and  the  corresponding  number  2-9094,  in  the  3d 
column,  multiplied  by  2691',  gives  7829  feet,  for  the  greatest 
range,  being  nearly  a mile  and  a half. 

Exam.  2.  In  like  manner,  the  same  balls  discharged  with 
the  velocity  860  feet,  will  have  for  its  greatest  range  3891 
feet,  or  nearly  f of  a mile,  and  the  elevation  producing  it 
39°  55'. 

These  examples,  and  indeed  the  whole  table  in  the  9th 

problem. 


Table  of  Elevations  giving  the 
Greatest  Range 


Initial  vel- 
div.  by  i;. 

Elevation. 

ir.ange  d;v. 
by  a 

0-6910 

44°  0/ 

0-3914 

0-9445 

43  15 

0-5850 

M980 

42  30 

0-7787 

1-4515 

41  45 

0-9724 

1-7050 

41  0 

1 1661 

1-9585 

40  15 

l-3c>98 

2-2120 

39  30 

1 5535 

2-4655 

38  45 

1-7472  ! 

2-7190 

38  0 

1-9409 

2-9725 

37  15 

2-1346 

3-2260 

36  30 

2 3283 

3-4795 

35  45 

2-5220 

3-7330 

35  0 

2-7157 

3-9865 

34  15 

2-9094 

4-2400 

33  30 

3-1031 

4-4935 

32  45 

3-2968 

4-7470 

32  0 

3-4905  j 

5-0000 

31  15  j 

3-6842 

476 


THEORY  AND  PRACTICE 


problem,  are  only  adapted  to  the  use  of  cannon  balls.  But 
i:  is  not  usual  and  indeed  not  easily  practicable,  to  discharge 
cannon  shot  at  such  elevations,  in  the  British  service,  that 
practice  being  the  peculiar  office  of  mortar  shells.  On  this 
account  then  it  will  be  necessary  to  make  out  a table  of  ter- 
minal velocities,  and  altitudes  due  to  them,  for  the  different 
sizes  of  such  shells.  The  several  kinds  of  these  in  present 
use,  are  denominated,  from  the  diameters  of  their  mortar 
bores  in  inches,  being  the  live  following,  viz.  the  4-6,  the 
5-8,  the  8,  tbe  10,  and  the  13  inch  mortars,  as  in  the  first 
column  of  the  following  table  But  the  outer  diameters  of 
the  shells  are  somewhat  smaller,  to  leave  a little  room  or 
space  as  windage,  as  contained  in  the  2d  column. 


Table  of  dimensions,  <$-e.  of  Mortar  Shells. 


D an:  of 
Morta.f. 

Diam.  of 
shells. 

W eight 
f Shells 
filled. 

VV  eigh 
of  equal 
solid. 

Ratio  of 

shell  t<> 

sol'd. 

Terminal 

velocity 

Alt.  a 

due  to 
veloc.  i 

i.Cii. 

inch. 

lbs. 

lb>. 

feet. 

feet. 

4-6 

4-63 

9 

12|- 

1-42 

314 

1541 

5 8 

-5-72 

18 

251 

1 42 

352 

1936 

8 

7-90 

47 

67 

1 42 

414 

2678 

10 

9 84 

9 1 A 

130 

1 42 

462 

3335 

13 

1 

12  80 

201 

286 

1 42 

527 

4340 

The  3d  coliynn  contains  the  weight  of  each  shell  when  the 
hollow  part  is  filled  with  powder  : the  diameter  of  the  hollow 
is  usually  T75  of  that  of  the  mortar  : the  weight  of  the  shells 
empty  and  when  filled,  with  other  circumstances,  maybe  seen 
at  quest  63,  in  Mensuration,  end  of  vol  1.  On  account  of  the 
vacuity  of  the  shell  being  filled  only  will)  the  gunpowder,  the 
weight  of  the  whole  so  filled,  and  contained  in  column  3,  is 
much  less  than  the  weight  of  the  same  size  of  solid  iron,  and 
the  corresponding  weights  of  such  equal  solid  balls  are  con- 
tained in  col.  4.  The  ratio  of  these  weights,  or  the  latter  di- 
vided by  the  former,  occupies  tbe  5th  column. 

Now  because  the  loaded  or  filled  shells  are  of  less  specific 
gravity,  or  less  heavy,  than  the  equal  solid  iron  halls,  in  the 
ratio  of  1 to  1-42,  as  in  co:  umn  5,  the  former  will  have  less 
power  or  force  to  oppose  the  resistance  ot  the  air,  in  that 
same  proportion,  and  the  terminal  or  greatest  velocity,  as 
determined  in  the  9th  prob.  will  be  correspondently  less. 
Therefore,  instead  of  the  rule  there  given,  viz.  175-5  y/  d,  for 

that  velocity,  the  rule  must  now  be  176'5^/j^-  = 147-3  v/rf=r< 

the 


OP  GUNNERY. 


477 


the  diameter  of  the  shell  being  d ; that  is,  the  terminal  velo- 
cities will  be  all  less  in  the  ratio  of  147-3  to  176  5.  Now, 
computing  these  several  velocities  by  this  rule,  to  all  the  dif- 
ferent diameters,  they  are  found  as  placed  in  the  6th  col.  ; 
and  in  the  7th  or  last  colump  are  set  the  altitude  which 
would  produce  these  velocities  in  vacuo,  as  computed  from 

this  theorem  ^7-. 

64 

Having  now  obtained  these  terminal  velocities,  and  their 
producing  altitudes,  for  the  shells,  we  can,  from  them  and 
the  former  table  of  ranges  and  elevations,  easily  compute  the 
greatest  range,  and  the  corresponding  angle  of  elevation,  for 
any  mortar  and  shell,  in  the  game  way  as  was  done  for  the 
balls  in  this  problem.  Thus,  for  example,  to  find  the  great- 
est range  and  elevation,  for  the  13  inch  shell,  when  projected 
with  the  velocity  of  2000  feet  per  second,  being  nearly  the 
greatest  velocity  that  balls  can  be  discharged  with.  Now, 

by  the  method  before  u-^ed  — 3-796  ; opposite  to  this, 

J 327 

found  in  the  first  column,  of  the  table  of  ranges,  corresponds 
34°49'  for  the  elevation  in  the  2d  column,  and  the  number 
2-764  in  the  3d  column  ; this  multiplied  by  the  altitude  4340, 
gives  1)995  feet,  or  more  than  2\  miles,  for  the  greatest 
range. 

This  however  is  much  short  of  the  distance  which  it  is  said 
the  French  have  lately  thrown  some  shells  at  the  siege  of 
Cadiz,  viz.  3 miles,  which  it  seems  has  been  effected  by 
i means  of  a peculiar  piece  of  ordnance,  and  by  loading  or  fill- 
1 ing  the  cavity  of  the  shell  with  lead  to  render  it  heavier,  and 
thus  make  it  fitter  to  overcome  the  resistance  of  the  air.  Let 
us  then  examine  what  will  be  the  greatest  range  of  our  13 
inch  shell,  if  its  usual  cavity  be  quite  filled  with  lead  when 
discharged,  with  the  projectile  velocity  of  2000  feet. 

Now  the  diameter  of  the  cavity,  being  about  of  that  of 
the  mortar  13,  will  be  nearly  9 inches.  And  the  weight  of 
a globe  of  lead  of  this  diameter  is  139-31!)  ; which  added  to 
187-8,  the  weight  of  the  shell  empty,  gives  3271b,  the  whole, 
weight  of  the  shell  when  the  cavity  is  filled  .with  lead,  which 
was  found  286  when  supposed  all  of  solid  iron,  their  ratio  or 
quotient  is  -8783.  1 hen,  as  before,  the  theorem  will  he 

175-5  y/  187  3y/d  for  the  terminal  velocity  ; which, 

when  d — 12-8,  becomes  670  for  the  terminal  velocity  : 

therefore  its  producing  altitude  is  = 7014.  Then  bv 

64  ’ 

the  same  method  as  before,  = 2-.9S5  ; which  number 

found 


478 


THEORY  AND  PRACTICE 


found  in  the  first  column  of  the  table  of  ranges,  the  opposite 
number  in  the  2d  col.  is  37°  15'  for  the  elevation  of  the  piece, 
and  in  the  3d  column  2T4,  multiplied  by  7014,  gives  15010 
feet,  or  nearly  3 miles.  So  that  our  13  inch  shells,  discharged 
at  an  elevation  of  about  37£  degrees,  would  range  nearly  the 
distance  mentioned  by  the  French,  when  filled  with  lead,  if 
they  can  be  projected  with  so  much  as  2000  feet  velocity,  or 
upwards.  This  however  it  is  thought  cannot  possibly  be 
effected  by  our  mortars  ; and  that  it  is  therefore  probable  the 
French,  to  give  such  a velocity  to  those  shells,  must  have 
contrived  some  new  kind  of  large  cannon  on  the  occasion. 

4.  Having  shown  in  the  preceding  articles  and  problems, 
how,  from  our  theory  of  the  air’s  resistance,  can  be  lound, 
first  the  initial  or  projectile  velocity  of  shot  and  shells  ; 2dly, 
the  terminal  velocity,  or  the  greatest  velocity  a ball  can  ac- 
quire by  descending  by  its  own  weight  in  the  air  ; 3dly,  the 
height  a ball  will  ascend  to  in  the  air,  being  projected  verti- 
cally with  a given  velocity,  also  the  time  of  that  ascent  ; 4thly, 
the  greatest  horizontal  ranges  of  given  shot,  projected  with  a 
given  velocity  ; as  also  the  particular  angle  of  elevation  of 
the  piece,  to  produce  that  greatest  range.  It  remains  then 
now  to  inquire,  what  laws  and  regulations  can  be  given  re- 
specting the  ranges,  and  times  of  flight,  of  projects  made  at 
other  angles  of  elevation. 

Relating  to  this  inquiry,  the  Encyclopaedia  Britannica 
mentions  the  two  following  rules  : 1st  “ Balls  of  equal  den- 
sity projected  with  the  same  elevation,  and  with  velocities 
which  are  as  the  square  roots  of  their  diameters,  will  describe 
similar  curves.  This  is  evident,  because  in  this  case,  the 
resistance  will  be  in  the  ratio  of  their  quantities  of  motion  ; 
therefore  all  the  homologous  lines  of  the  motion  will  be  in 
the  proportion  of  the  diameters.”  But  though  this  may  be 
nearly  correct,  yet  it  can  hardly  ever  be  of  any  use  in  prac- 
tice, since  it  is  usual  and  proper  to  project  small  balls,  not 
with  a less,  but  with  a greater  velocity,  than  the  larger  ones. 
2dly,  the  other  rule  is,  “ If  the  initial  velocities  of  balls,  pro- 
jected with  the  same  elevation,  be  in  the  in-verse  subduplicate 
ratio  of  the  whole  resistance,  the  ranges,  and  all  the  homo- 
logous lines  in  their  track,  will  be  inversely  as  those  resist- 
ances.” This  rule  will  come  to  the  same  thing,  as  having 
the  initial  velocities  in  the  inverse  ratio  of  the  diameters,  as 
distant  perhaps  from  fitness  as  the  former.  Two  tables  are 
next  given  iu  the  same  place,  for  the  comparison  of  ranges 
and  projectile  velocities,  the  numbers  in  which  appear  to  be 
much  wide  of  the  truth,  as  depending  on  very  erroneous 
effects  of  the  resistance.  Most  of  the  accompanying  remarks, 

however, 


OF  GUNNERY. 


479 


however,  are  very  ingenious,  judicious,  and  philosophical, 
and  very  justly  recommending  the  making  and  recording  of 
good  experiments  on  the  ranges  and  times  of  flight  of  pro- 
jects, of  various  sizes,  made  with  different  velocities,  and  at 
various  angles  of  elevation. 

Besides  the  above,  we  find  rules  laid  down  by  Mr.  Robins 
and  Mr.  Simpson,  for  computing  the  circumstances  relating 
to  projectiles  as  affected  by  the  resistance  of  the  air.  Those 
of  the  former  respectable  author,  in  his  ingenious  Tracts  on 
Gunnery,  being  founded  on  a quantity  which  he  calls  f, 
(answering  to  our  letter  a in  the  foregoing  pages),  I find  to  be 
almost  uniformly  double  of  what  it  ought  to  be,  owing  to  his 
improper  measures  of  the  air’s  resistance  ; and  therefore  the 
conclusions  derived  by  means  of  those  rules  must  needs  be 
very  erroneous.  Those  of  the  very  ingenious  Mr.  Simpson, 
contained  in  his  Select  Exercises,  being  partly  founded  on 
experiment,  may  bring  out  conclusions  in  some  of  the  cases 
not  very  incorrect  ; while  some  of  them  particularly  those 
relating  to  the  impetus  and  the  time  of  flight,  must  be  very 
wide  of  the  truth.  We  must  therefore  refer  the  student, 
for  more  satisfaction,  to  our  rules  and  examples  before  given 
in  pa.  142  this  vol.  &c  especially  for  the  circumstances  of  dif- 
ferent ranges  and  elevations,  &c.  after  having  determined,  as 
above,  those  for  the  greatest  ranges,  founded  on  the  real  mea- 
sure of  the  resistances. 


PROMISCUOUS 


[ 480  ] 


PROMISCUOUS  PROBLEMS,  AS  EXERCISES  IN  MECHANICS, 
STATICS,  DYNAMICS,  HYDROSTATICS,  HYDRAULICS,  PRO. 
4ECTILES,  kc.  kc. 


PROBLEM  L 


Let  ab  and  ac  be  two  inclined  planes,  whose  common  altitude 
ad  is  given  = 64  feet : and  their  lengths  such,  that  a heavy  body 
is  2 seconds  of  time  longer  in  descending  through  ae  than  through 
ac,  by  the  force  of  gravity  ; and  if  two  balls,  the  one  weighing 
3 and  the  other  2/6,  be  connected  by  a thread  and  laid  on  the 
planes,  the  thread  sliding  freely  over  the  vertex  a,  they  will  mu- 
tually sustain  each  other.  Quere  the  lengths  of  the  two  planes. 

The  lengths  of  the  planes  of  the  same  height  being  as  the 
times  of  descent  down  them  (art  133  this  vol.),  and  also  as  the 
weights  of  bodies  mutually  sustaining  each  other  on  them 
(art.  122),  therefore  the  times  must  be  as  the  weights  ; hence 
as  1,  the  difference  of  the  weights,  is  to  2 sec.  the  difl’.  of 

times,  : : < ? . Sec"  > the  times  of  descending  down  the  two 

^2:4  sec.  ^ ° 

planes.  And  as  y'  16  : ^/  64  : : 1 sec.  : 2 sec.  the  time  of  de- 
scent down  the  perpendicular  height  (art  70,).  Then  by  the 

laws  of  descents  (art.  132),  as  2 sec.  : 64  feet  | ^ | | | 

feet,  the  lengths  of  the  planes. 


Note.  In  this  solution  we  have  considered  16  feet  as  the 
space  freely  desended  by  bodies  in  the  ist  second  of  time, 
and  32  feet  as  the  velocity  acquired  in  that  time,  omitting 
the  fractions  Tc  and  to  render  the  numeral  calculations 
simpler  as  was  done  in  the  preceding  chapter  od  projectiles, 
and  as  w.e  shall  do  also  in  solving  the  following  questions, 
Wherever  such  numbers  occur. 

Another  Solutioti  by  means  of  Algebra. 

put  x = the  time  of  descent  down  the  less  plane  ; then 
will  x + 2 be  that  of  the  greater,  by  the  question.  Now 
the  weights  being  as  the  lengths  of  the  planes,  and  these 
again  as  the  times,  therefore  as  2 : 3 : : sc  : at  + 2 ; hence 

2.r  + 


PROMISCUOUS  EXERCISES. 


481 


2.r  -f-  4 =»  3x,  and  x = 4 sec.  Then  the  lengths  of  the 
planes  are  found  as  in  the  last  proportion  of  the  former  solu- 
tion. 

PROBLEM  2. 

If  an  elastic  ball  fall  from,  the  height  of  50  feet  above  the 
plane  of  the  horizon , and  impinge  on  the  hard  surface  of  a 
plane  inclined  to  it  in  an  angle  of  15  degrees  ; it  is  required,  to 
find  what  part  of  the  plane  it  must  strike , so  that  after  reflection, 
it  may  fall  on  the  horizontal  plane,  at  the  greatest  distance  possi- 
ble beyond  the  bottom  of  the  inclined  plane  ? 

Here  it  is  manifest  that 
the  ball  must  strike  the  ob- 
lique plane  continued  on  a 
point  somewhere  below  the 
horizontal  plane  ; for  other- 
wise there  could  be  no  maxi- 
mum. Therefore  let  bc  be 
the  inclined  plane,  cdg  the  horizontal  one.  b the  point  on 
! which  the  ball  impinges  after  falling  from  the  point  a,  begi 
the  parabolic  path,  e its  vertex,  bh  a tangent  at  b,  being  the 
direction  in  which  the  ball  is  reflected  ; and  the  other  lines  as 
are  evident  in.  the  figure.  Now,  by  the  laws  of  reflection, 
the  angle  of  incidence  abc,  is  equal  to  the  angle  of  reflection 
hbm,  and  therefore  this  latter,  as  well  as  the  former,  is  equal 
to  the  complement  of  the  the  inclination  of  the  two  planes  ; 
but  the  part  ibm  is  = j/c,  therefore  the  angle  of  projection 
hbi  is  = the  comp,  of  double  the  ^c,  and  being  the  comp,  of 
hbk,  theref.  ^/hbk  = 2 ^c.  Now,  put  a = 50  = ad  the 
height  above  the  horizontal  line,  t = tang.  Zdbc  or  75°  the 
complement  of  the  plane’s  inclination,  r = tang,  hbi  or  g£H 
= 80°  the  comp,  of  2 ^c,  s = sine  of  2 ^hbi  = 120°  the 
double  elevation,  or  = sine  of  4^c;  also  x — ab  the  impe- 
tus or  height  fallen  through.  Then, 

bi  = 4kh  = 2sx,  by  the  projectiles  prop.  21, 

, 4 bk  = r X kh  = ±srx  ) , . . 

and  l CD  = t X bd  = i (x-a).  $ b?  trigonometry  ; 

also,  kd  = ek  — bd  = i stx  — x + a,  and  ke  = Abi  — sx  ; 
then,  by  the  parabola,  bk  : dk  : : ke  :'fg  = ke  X 

ax  — (—  — s2  ) x2  ] = 

v T 

26  [ax  — 62x2),  putting  6 = sine  of  2 g^c  = sine  of  30°. 

Hence  cg  — cd  -f*  df  ± fg  = tx  — ta  -j-  sx  ± 26  [ax—b2x2) 

a maximum,  the  fluxion  of  which  made  = 0,  and  the  equa- 
tion reduced,  gives  x —~x  (1  ±:  */, — — , where n = s 

2A2  V V(?l3-J-464) 


KD  rs^X-  — ‘isx-  -j-  2asx  _ 

V — = v' = v/[ 

V KD  T 


'Vot.  II. 


62 


f-.t. 


482 


P&PMISCUOUS  EXERCISES. 


4"  t,  and  the  double  sign  ± answers  to  the  two  roots  or  va- 
lues of  x,  or  to  the  two  points  g,  g,  where  the  parabolic  path 
cuts  the  horizontal  line  co,  the  one  in  ascending  and  the  other 
in  descending. 

Now,  in  the  present  case,  when  the  /_c  — 15°,  t = tang. 
75°  = 2 -f  3,  r = tan  bO°=v/  3,  s = sin.  60°=^  3,  b — 

sin.  30°  = -i,  n = s + t = 2-f-f  . / 3;  then  — = 2a  = 100,  and 

“ Zb2 

n-  71  41-6^3  . a , r.2 

«2  q-4  t>i  n2-e}  52  -'02  v vn*44c4' 

= 100  X (1  ± i v'1— 3— ) = 1 0°  X ( 1 ± - 9941 4)  = 199-,14 

or  *586  ; but  the  former  must  be  taken.  Fence  the  body 
must  strike  the  inclined  plane  at  149-414  feet  below  the  ho- 
rizontal line  ; and  its  path  after  reflection  will  cut  the  said 

line  in  two  points  ; or  it  will  touch  it  when  x = Hence 

ho 

also  the  greatest  distance  cg  required  is  826  9915  feet 

Carol.  If  it  were  required  to  find  cg  or  tx-  ta  4*  sx  — 
2 by/(ax  — b2 x2)  — g a given  quantity,  this  equation  would 
give  the  value  of  x by  solving  a quadratic. 

PROBLEM  3. 


Suppose  a ship  to  sail  from  the  Orkney  Islands , in  latitude 
59°  3'  north , on  a n . s.  e.  course,  at  the  rate  of  10  miles  an 
hour  ; it  is  required  to  determine  how  long  it  will  be  before  she 
arrives  at  the  pole,  the  distance  she  will  have  sailed,  and  the  dif- 
ference of  longitude  she  will  have  made  when  she  arrives  there  ? 

Let  abc  represent  part  of  the  equator  ; 
p the  pole  ; Amrp  a loxodromic  or  rhumb 
line,  or  the  path  of  the  ship  continued  to  the 
equator  ; pb,  pc,  any  two  meridians  indefi- 
nitely near  each  other  ; nr,  or  mt,  the  part 
of  a parallel  of  latitude  intercepted  between 
them. 

Put  c for  the  cosine,  and  t for  the  tangent 
of  the  courge,  or  angle  nmr  to  the  radius  r ; 

Am,  any  variable  part  of  the  rhumb  from  the  equator,  = v j 
the  latitude  e in  = w ; its  sine  x,  and  cosine  y ; and  ab,  the 
dif.  of  longitude  from  a,  = ?.  Then,  since  the  elementary 
triangle  m nr  may  be  considered  as  a right-angled  plane  tri- 
angle, it  is,  as  rad.  r : c — sin.  £mrn  : : 4 = mr  : w = mn 

: : v : w ; theref.  cv  = rw,  or  v = — = — , by  putting  s for 

the  secant  of  the  ,/nmr  the  ship’s  course.  In  like  man- 
ner, 


PROMISCUOUS  EXERCISES. 


483 


iier,  if  w be  any  other  latitude,  and  vits  corresponding  length 

,w-  v> 


pf  the  rhumb  ; 


, r w 

then  v = — 

c 


and  hence  v—v=  r X 


or  d ='— , by  putting  o ■=  v — v the  distance,  and  d — w — -ai 

the  dif  of  latitude  ; which  is  the  common  rule. 

The  same  is  evident  without  fluxions  : for  since  the  £ mrn 
is  the  same  m whatever  point  of  the  path  Amrp  the  point  m 
is  taken,  each  indefinitely  small  particle  of  a-/«?-p,  must  be  to 
the  corresponding  indefinitely  small  part  of  b in,  in  the  con- 
stant ratio  of  radius  to  tiie  cosine  of  the  course  : and  there- 
fore the  whole  lines,  or  any  corresponding  parts  of  them,  must 
be  in  the  same  ratio  also,  as  above  determined  In  the  same 
maimer  it  is  proved  that  radius  : sine  of  the  course  : : dis- 
tance : the  departure. 

Again,  as  the  radius  r : t = tang  mnr  : : w — mn  : nr  or  mt, 
and  as  r : y : : pb  : Pin  : : z — bc  : mt  ; hence,  as  the  extremes 
of  these  proportions  are  the  same,  the  rectangles  of  the  means 

must  be  equal,  viz.  yz  = tw  — — because  w — — by  the 

property  of  the  circle  ; theref.  z = — = i the  general 

fluents  of  these  are  z — t X hyp  l°S-v/~ ^ + c ; which 
corrected  by  supposing  z — 0 whenx  = a,  are  z —t  X (hyp. 

!°g-  y/[~x  - hyp-  lo» v~) ; but r x (fayp-  log-  v rr+-x 

— hyp.  log.  is  the  meridional  parts  of  the  dif  of  the 

latitudes  whose  sines  are  x and  a,  which  call  b : then  is 


to 


, the  same  as  it  is  by  Mercator’s  sailing. 


Further,  putting  m — 2-71828  the  number  whose  hyp.  log. 

is  1,  and,  n — \ then,  when  z begins  at  a,  mn  =r-^  an(j 

t r — x 

71  ^ T 

theref.  x = r X — r : hence  it  appears  that 

mn  i- 1 »B«+  1 1 

as  m",  or  rather  »orj  increases  (smce  m is  constant),  that  x 

2r 

approximates  to  an  equality  with  r,  because^  ^ decreases  or 

converges  to  0,  which  is  its  limit  ; consequently  r is  the 
limit  or  ultimate  value  of  x : but  when  x — r.  the  ship  will 
be  at  the  pole  ; theref.  the  pole  must  be  the  limit,  or  eva- 
nescent -tate,  of  the  rhfumh  or  coarse  : so  that  the  ship  may 
be  said  to  arrive  at  the  pole  after  making  an  infinite  number 

of  revolutions  round  it  ; for  the  above  expression  van- 

ishes 


484 


PROMISCUOUS  EXERCISES. 


ishes  when  n,  and  consequently  z,  is  infinite,  in  which  case  x 
is  — r. 

Now,  from  the  equation  d = — = — , it  is  found,  that 

c r 

when  d = 30°  57',  the  comp,  of  the  given  lat.  59°  3',  and  c = 
sine  of  67°,  30'  the  comp  of  the  course,  n will  be  = 2010 
geographical  miles,  the  required  ultimate  distance  : which, 
at  the  rate  of  10  miles  an  hour,  will  be  passed  over  in  201 
hours,  or  8f  days.  The  dif.  of  long  is  shown  above  to  be  in- 
finite When  the  ship  has  made  one  revolution,  she  will  be 
but  about  a yard  from  the  pole,  considering  her  as  a poiut 

When  the  ship  has  arrived  infinitely  near  the  j ole,  she  will 
go  round  in  the  manner  of  a top,  with  an  infinite  velocity  ; 
which  at  once  accounts  for  this  paradox,  viz.  thdt  though  she 
make  an  infinite  number  of  revolutions  round  the  pole,  yet 
her  distance  run  will  have  an  ultimate  and  definite  value,  as 
above  determined  : for  it  is  evident  that  however  great  the  num- 
ber of  revolutions  of  a top  may  be,  the  space  passed  over  by 
its  pivot  or  bottom  point,  while  it  continues  on  or  nearly  on  the 
same  point,  must  be  infinitely  small  or  less  than  a certain  as- 
signable quantity. 


PROBLEM  4. 


A current  of  water  is  discharged  by  three  equal  openings 
or  sluices,  in  the  following  shapes  : the  first  a rectangle,  the 
second  a semicircle,  and  the  third  a parabola,  having  their 
altitudes  equal  and  their  bases  in  the  same  horizontal  line,  and 
the  water  level  with  the  tops  of  the  arches  : on  this  supposition 
it  is  required  to  show  what  may  be  the  proportion  of  the  quantities 
discharger!  by  these  sluices. 

Let  vb  be  half  the  parallelogram,  avc 
half  the  semicircle,  and  avd  half  the  pa- 
rabola. that  is,  the  halves  of  the  respective 
sluices  or  gates  Put  a = av  the  common 
altitude,  and  c = 7851  : then  is  co2  the 
area  of  each  of  the  figures  ; also  ca  — ab, 
a — ac,  and  \ca  — ad  ; also  put  x — vr 
any  variable  depth,  and  x = pp.  Then,  the  water  discharged, 
at  any  depth  x,  being  as  the  velocity  and  aperture,  and  the 
velocity  being  in  all  the  figures  as  y x,  therefore  x y'xX  fq, 

and  x-Jx  X PR,  and  f v'  x X rs,  or  cax^x,  andzj^y^a — x), 
and  fcyaX  *x,  are  proportional  to  the  fluxions  of  the  quan- 
tity of  water  discharged  by  the  said  figures  or  sluices  re- 
spectively ; the  correct  fluents  of  which,  when  x — a,  are 

|ca- , and  T%a2  (8  y”  2 — 7),  and  fca2,  the  2d  fluent  being 
found  by  art.  60  pa.  336  of  this  vol.  Hence  the  quantities 

of 


PROMISCUOUS  EXERCISES. 


485 


of  water  discharged  by  the  rectangle,  the  semicircle,  and  the 
parabola,  are  respectively  as  f c,  and  T2T(8  2 — 7),andfc,  or 

as  1,  and  ^(8^/2 — 7),  andf,  or  as  1,  and  1-09847,  and  If. 

PROBLEM  5. 

The  initial  velocity  of  a 24 lb  ball  of  cast  iron,  -which  is  pro - 
jected  in  a direction  perpendicular  to  the  horizon,  being  sup- 
posed 1200  feet  per  second  ; and  that  the  resistance  of  the  me- 
dium is  constantly  as  the  square  of  the  velocity,  and  every  where 
of  the  same  density  : required  the  time  of  flight,  and  the  height 
to  which  it  will  ascend 

Answer.  By  problems  5 and  6,  of  the  last  chapter,  the 
ascent  will  be  found  = 5337  feet  and  the  time  of  the  ascent 
28  seconds. 

PROBLEM  6. 

To  determine  the  same  as  in  the  last  question,  supposing  the 
i density  of  the  atmosphere  to  decrease  in  ascending  after  the  usual 
way  ? 

Ans.  By  probs.  7 and  8,  the  height  will  be  5614  feet,  and 
the  time  34  seconds. 

PROBLEM  7. 

It  is  required  to  find  the  diameter  of  a circular  parachute,  by 
means  of  which  a man  of  150 Ih  weight  may  descend  on  the  earth , 
from  a balloon  at  a height  in  the  air,  with  the  velocity  of  only  10 
feet  in  a second  of  time,  being  the  velocity  acquired  by  a body 
freely  descending  through  a space  of  only  1 foot  6f  inches,  or 
of  a man  jumping  down  from  a height  of  18f  inches  : the  para- 
! chute  being  made  of  such  materials  and  thickness,  that  a circle  of 
it  of  50  feet  diameter,  weighs  only  15016,  and  so  in  proportion 
more  or  less  according  to  the  area  of  the  circle. 

If  a falling  body  descend  with  a uniform  velocity,  it  must 
necessarily  meet  with  a resistance,  from  the  medium  it  de- 
scends in,  equal  to  the  whole  weight  that  descends.  Let  x 
denote  the  diameter  of  the  parachute,  and  a — -7854  ; then 
ax2  will  be  its  area,  and  as  502  : x2  : : 150  : fgX2  the  weight 
of  the  same,  to  which  adding  150lb,  the  man’s  weight,  the 
sum  -g\x2  +150  will  be  the  whole  descending  weight.  Again, 
in  the  table  of  resistances  (in  the  scholium  to  prop.  22,  mol.  of 
bod.  in  Fluids),  we  find  that  a circle  off  of  a square  foot  area, 
moving  with  10  feet  velocity,  meets  with  a resistance  of  -57 
ounces  = -0475  lb;  and  the  resistances,  with  the  same  velocity, 

being 


486 


PROMISCUOUS  EXERCISES. 


being  as  the  surfaces,  therefore,  as  | : -0475  : : axa  : -21375axs 
= • 16788a:2  the  resistance  of  the  air  to  the  parachute,  to  which 
the  descending  weight  must  be  equal  ; that  is,  -16788a:2  = 
+ 150  ; hence  -10788a:2  = 150,  or  x 2 = 1390-5,  and 
hence  x = 37f  feet,  the  diameter  of  the  parachute  required. 

PROBLEM  8. 

To  determine  the  effects  of  Pile-Engines. 

The  form  of  the  pile-engine,  as  used  by  the  ancients,  is  not 
known  Many  have  been  invented  and  described  by  the 
moderns.  Among  all  these,  that  appears  to  be  the  best  which 
was  invented  by  Vauloue,  as  desciibed  by  Desaguliers,  and 
was  used  at  piling  the  foundations  at  building  Westminster 
Bridge.  Its  chief  properties  are,  that  the  ram  or  weight  be 
raised  with  the  least  expence  of  force,  or  with  the  fewest 
men  ; that  it  fall  freely  from  its  greatest  height ; and  that, 
having  fallen,  it  is  presently  laid  hold  of  by  the  forceps,  and 
so  raised  up  to  its  height  again.  By  which  means,  in  the 
shortest  time,  and  with  the  fewest  men,  or  the  least  force,  the 
most  piles  can  be  driven  to  the  greatest  depth. 

Belidor  has  given  some  theory  as  to  the  effect  of  the  pile- 
engine  but  it  appears  to  be  founded  on  an  erroneous  prin- 
ciple : he  deduces  it  from  the  laws  of  the  collision  of  bodies. 
But  who  does  not  perceive  that  the  rules  of  collision  suppose 
a free  motion  and  a r.on-resisting  medium  ? It  cannot  there- 
fore be  applied  in  the  present  case,  where  a very  great  re- 
sistance is  opposed  to  the  pile  by  the  ground.  We  shall 
therefore  here  endeavour  to  explain  another  theory  of  this 
machine. 

Since  the  percussion  of  the  weight  acts  on  the  pile  during 
the  whole  time  the  pile  is  penetrating  and  sinking  in  the 
earth,  by  each  blow  of  the  ram.  during  which  time  its  whole 
force  is  spent  ; it  is  manifest  that  the  effect  of  the  blow  is  of 
that  nature  which  requires  the  force  of  the  blow  to  be  esti- 
mated by  the  square  of  the  velocity.  But  the  square  of  the 
velocity  acquired  by  the  fall  of  the  ram,  is  as  the  height  it 
falls  from  ; therefore  the  force  of  any  blow  will  be  as  the 
height  fallen  through.  But  it  is  also  more  or  less  in  propor- 
tion to  the  weight  bf  the  ram  ; consequently  the  effect  or 
force  of  each  blow  must  be  directly  in  the  compound  ratio  of 
both,  viz.  as  an.',  where  w denotes  the  weight,  and  a the  alti- 
tude it  falls  from  ; or  it  will  be  simply  as  the  altitude  a,  when 
the  weight  w is  constant. 

Again,  the  force  of  the  blow  is  opposed  by  the  mass  of  the 
pile,  and  by  the  consistence  of  the  earth  penetrated  by  the 

point 


PROMISCUOUS  EXERCISES. 


487 


point  of  the  pile,  and  also  by  the  friction  of  the  earth  against 
the  surface  or  sides  of  the  pile  that  have  penetrated  below  the 
surface.  Consequently  the  effect  of  the  blow,  or  the  depth 
penetrated  by  the  pile,  will  be  inversely  in  the  compound 
ratio  of  these  three,  viz,  inversely  as  mtf,  where  m denotes 
the  mas*  of  the  pile,  t th-'  tenacity  or  cohesion  of  the  earth, 
and  f the  friction  of  the  surface  penetrated  in  the  earth.  But, 
in  the  same  soil  and  with  the  same  pile,  in  and  t are  both 
constant,  in  which  case  the  depth  of  penetration  will  be  in- 
versely only  as  / the  friction.  On  all  accounts  then  the  pe- 
netration will  be  as  or  simply  as  ~ only,  for  the  same 
mtf  1 f 

weight  and  pile  and  soil. 

To  determine  the  depth  sunk  by  the  pile  at  each  stroke  of  the  ram. 

After  a few  strokes,  so  as  to  give  the  pile  a little  hold  in 
the  ground,  to  make  it  stand  firmly,  the  blows  of  the  ram 
may  be  considered  as  commencing  and  causing  the  pile  to 
sink  a little  at  every  stroke,  by  which  small  successive  sink- 
ings of  the  pile,  the  space  the  ram  falls  through  will  be  suc- 
cessively increased  by  these  small  accessions,  and  the  force  of 
the  successive  blows  proportionally  increased,  But  these,  on 
the  other  hand,  are  resisted  and  opposed  by  the  friction  of 
the  part  of  the  pile  which  has  been  sunk  before,  and  which 
also  sinks  at  each  stroke  ; and  as  the  quantities  of  these  rub- 
bing surfaces  increase  in  a greater  ratio  to  each  other,  than 
the  heights  fallen  through,  that  is,  the  resisting  forces  in- 
creasing faster  than  the  impelling  forces,  it  is  manifest  that 
the  depths  successively  sunk  by  the  blows  must  gradually 
decrease  by  little  and  little  every  time  ; which  is  also  found 
to  be  quite  conformable  to  experience.  Thus  then  the  suc- 
cessive sinkings  will  proceed  gradually  diminishing,  till  they 
become  so  small  as  to  be  almost  imperceptible. 

Nowit  was  found  above  that  ^.is  as  the  penetration  by 

any  blow  of  the  ram,  by  the  same  pile  in  the  same  soil,  that  is, 
as  the  height  fallen  directly,  and  as  the  resistance  or  friction 
in  the  earth  inversely.  Let  a denote  any  other  and  greater 
height,  by  an  after  stroke,  and  f its  friction  ; also  pthe  pene- 
tration by  the  former  blow,  and  p that  by  the  latter,  which 
must  be  the  smaller  : then,  by  the  foregoing  principle, 


theorem 


r : p ; hence  « : a : : /p  : f p,  which  is  a general 


But 


483 


PROMISCUOUS  EXERCISES. 


But  now,  with  respect  to  the  quantity  of  friction  from  any 
blow,  though  it  be  not  known  from  experiment  that  the  fric- 
tion is  exactly  proportional  to  the  rubbing  surface,  there  is 
great  reason  to  believe  that  it  must  be  at  least  very  nearly 
so  : there  is  also  equal  reason  to  conclude  that  the  effect 
or  resistance  from  that  rubbing  surface  must  be  nearly 
or  exactly  as  the  length  of  space  it  moves  over,  that  is  by  the 
penetration  of  the  pile  by  any  blow.  Now,  if  d denote  the 
depth  of  the  pile  in  the  ground  before  any  new  blow  is  struck 
by  the  ram,  and  b the  depth  or  penetration  produced  by  the 
blow,  then  the  length  of  the  rubbing  surface  will  b e d-f 
for,  the  length  ef  the  rubbing  surface  is  only  d at  the  begin- 
ning of  the  motion,  and  it  is  d + 6 at  the  end  of  it,  the  me- 
dium of  the  two,  or  d + \ b,  is  therefore  the  due  length  of 
the  surface,  and  the  space  or  depth  it  moves  over  is  6 ; there- 
fore the  whole  resistance  from  the  friction  is  (d  + i 6)  b If 
d then  denote  any  other  depth  of  the  pile  in  the  earth,  and 
b'  the  next  penetration,  then  (d  -j-  b ) V will  be  its  friction. 
Substituting  now  b for  p,  and  o'  for  p,  also  d -f-  a b for  /,  and 
d -f-  i b'  for  f,  in  the  general  theorem  a : a : : /p  : f p,  it  be- 
comes a : a : : {d  + \b)b  : (d -\-±b')b' , for  the  general  relation, 
between  the  heights  fallen  and  the  resistance  and  penetration. 

This  theorem  will  very  conveniently  give  the  series  of  ef- 
fects, or  successive  sinkings  of  the  piles,  by  the  blows  of  the 
ram.  Thus,  after  the  pile  has  been  properly  tixed,  or  indeed 
driven  to  any  depth  in  the  earth,  denoted  by  d,  then  to  give 
a blow,  the  ram  falls  from  the  height  a + d,  and  thereby 
sinks  the  pile  the  space  b suppose  ; hence  for  the  next  stroke, 
the  fall  will  bea+d+6  = a in  the  theorem  above,  and 
d bb  — d + 6 -f-  i&',  the  next  penetration  or  sinking  being 
b'  ; theref.  a + d : a+ d + b i : (d  + ±b)i  : (d  -f  b+±b)'b't 
a proportion  which  gives  the  quadratic  equa.  fc'a+26  (d-j-6)  = 


parison  with  a -f-  d. 

Now,  for  an  example  in  numbers,  suppose  a = 5 feet  = 
60  inches,  d = 10,  b =3,  that  is  a =60  the  height  of  the 
ram  above  the  top  of  the  pile  before  this  enters  the  ground  ; 
d — 10,  after  being  fixed  in  the  ground  ; and  b — 3 the 


a+t/-l-6 


X (2 d -f-  b)b , the  root  of  which  is  b ' = — (d-j-6)  + 


PROMISCUOUS  EXERCISES, 


48S 


the  2d  stroke.  Next,  substituting 
d-{-b  for  d,  and  b'  for  b,  the  same  . 
theorem  gives  24-8  for  the  next 
sinking,  or  the  next  value  of  b'. 
And  so  on  continually,  by  which 
means  the  series  of  the  successive 
corresponding  values  of  the  letters 
will  be  as  in- the  margin,  the  Iasi 
column  showing  the  several  suc- 
cessive sinkings,  of  the  pile  by  the 
repeated  strokes  of  the  ram. 


Scholium.  Thus  then  it  appears  that  the  effect  of  any  ope- 
ration of  pile-driving  may  be  determined-  It  is  manifest  also 
that  the  greater  a is,  or  the  higher  the  top  of  the  machine  is 
where  the  ram  falls  from,  above  the  top  of  the  pile  at  first, 
i the  greater  will  be  every  stroke  of  the  ram,  and  consequently 
the  fewer  the  strokes  requisite  to  drive  the  pile  to  the  requi- 
site depth.  But  then  every  stroke  will  take  a longer  time, 
as  the  ram  will  be  both  longer  in  falling  and  longer  in  rais- 
ing : so  that  it  may  be  a question  whether  on  the  whole  the 
business  may  be  affected  in  the  less  time  by  a greater  height 
ef  the  machine,  or  whether  there  be  any  limit  to  the  height, 
so  as  to  produce  the  greatest  effect  in  a given  time. 

To  answer  this  question,  let  x denote  the  indeterminate 
height  from  which  any  weight  w is  to  fall,  z the  time  of  rais- 
ing it  after  a fall,  which  time  is  supposed  to  be  as  the  height 
x to  which  it  is  raised,  also  m the  given  time  of  producing  a 
proposed  effect  ; then  i^i  = the  time  of  the  weight  fall- 
ing ; therefore  % x z = the  whole  time  of  one  stroke  ; 

Conseq.  — m — or — — is  the  number  of  strokes  made  in 

-/X- (-43 

the  given  time  m,  and  hence f—  = the  whole  force  or 

a • v/x-f-4  2 

effect  in  the  time  m.  Now  this  effect  or  fraction  increases  con- 
tinually as  a; -increases,  because  the  numerator  increases  faster 
than  the  denominator,  since  the  former  increases  as  x,  while 
in  the  latter  though  the  one  term  z increases  as  x,  yet  the 
other  term  x only  increases  as  the  root  of  x.  So  that,  on 
the  whole,  it  appears  that  the  effect,  in  any  given  time,  in- 
creases more  and  more  as  the  height  is  increased. 


Specimen  of  the  Se- 
ries of  the  Successive 
values  of  d , b,  b. 

d 

b 

b' 

10 

3 

2-65 

13 

265 

2-48 

15-6S 

2-49 

2 32 

18-14 

2-32 

2 19 

20-46 

&c. 

2-19 

208 

Vol.  IP. 


63 


490  PROMISCUOUS  EXERCISES. 

PROBLEM  2. 

To  determine  how  far  a man , who  pushes  with  the  force  oj 
100 lb,  can  force  .a  sponge  into  a piece  of  ordnance,  whose  di- 
ameter is  5 inches,  and  length  ten  feet,  when  the  barometer  stands 
at  30  inches : the  vent,  or  touch-hole,  being  stopped,  and  the 
sponge  having  no  windage,  that  is,  fitting  the  bore,  quite  close  ? 

A column  of  quicksilver  30  inches  high,  and  5 in  diameter, 
is  52  X 30  X -7854  — 589-05  inches  ; which,  at  8-102  oz. 
each  inch,  weighs  4172-48  oz  or  298-281b,  which  is  the  pres- 
sure of  the  atmosphere  alone,  being  equal  to  the  elasticity  of 
the  ail  in  its  natural  state  ; to  this  addiug  the  1001b,  gives  j 
.3S8-281b,  the  whole  external  pressure.  Then,  as  the  spaces 
which  a quantity  of  air  possesses,  under  different  pressures, 
are  in  the  reciprocal  ratio  of  those  pressures,  it  will  be,  as 
398-28  : 290-23  ; : 10  feet  or  120  inches  : 90  inches  nearly,  ( 
the  space  occupied  by  the  air  ; theref.  120—90  = 30  inches, 
is  the  distance  sought. 

PROBLEM  10. 

To  assign  the  Cause  of  the  Deflection  of  Military  Projectiles, 

It  having  been  surmized  that  in  the  practice  of  artillery, 
the  deflexion  of  the  shot  in  its  flight,  to  the  right  or  left,  from 
the  ime  or  direction  the  gun  is  laid  in,  chiefly  arises  from  the 
moti'  a of  the  guu  during  the  time  the  shot  is  passing  out  of 
the  piece  •,  it  is  required  to  determine  what  space  an  18  pound- 
er will  recoil  or  fly  back,  while  the  shot  is  passing  out  of  the 
gun  ; supposing  its  weight  to  be  48001b  tbat  of  the  carriage 
24C0lb,  the  quantity  of  powder  81b,  tbe  length  of  the  cylinder 
108  inches,  that  of- the  charge  13  inches,  and  the  diameter  of 
the  bore  5-13  inches  ; supposing  aiso  that  the  resistance  from 
the  friction  between  the  platform  and  carriage  is  equal  to 
36001b  ? 

It  is  well  known  that  confined  gunpowder,  when  fired, 
immediately  changes  in  a great  measure  into  an  elastic  air, 
which  endeavours  to  expand  in  all  directions.  Now,  in  the 
question,  the  action  of  this  fluid  is  exerted  equally  on  the 
bottom  of  the  bore  of  tbe  gun  and  on  the  ball,  during  the 
passage  of  the  latter  through  the  cylinder  ; the  two  bodies 
therefore  move  in  opposite  directions,  with  velocities  which 
are  at  ail  times  in  the  inverse  ratio  of  the  quantities  of  matter 
moved.  Now  leti  .be  the  space  through  which  the  gun  re- 
coils ; then,  as  the  charge  occupies  13  inches  of  the  barrel, 
and  the  semidiameter  of  the  barrel  is  2-565,  the  space  moved 

through 


PROMISCUOUS  EXERCISES. 


401 


through  by  the  ball  when  it  quits  the  piece,  is  108  — < 13  — 
2-565  - x — 92-435  — x : and  as  the  elastic  fluid  expands 
in  both  directions,  the  quantity  whim  advances  towards  the 
muzzle,  is  to  that  which  retreats  from  it,  as  32-435  — x to  x : 


conseq. 


8x 


and 


92-43.->  -=•  x 


, „„  X-  8 are  the  quantities  of  the 

92-435  .92-435  1 

powder  which  move,  the  former  with  'he  gun,  and  the  latter 
with  the  ball  ; besides  these,  the  weight  of  ball  that  moves 
forwards  being  181b,  and  of  the  weights  and  resistance  back- 
wards 4800  -j-  2400  + 3600  = 108001b,  hence  the  whole 

8-37 

weights  moved  in  the  two  directions  are  10800  -f-  and 

•,0  , 92-435  - x 0 993298+8*  ,2403-31  — 8*  ,, 

18  + — X 8,  or  „ — and  — , or  as  the 

92-435  ■ 92435.  92-435 

numerators  of  these  only.  But  when  the  time  and  moving 
force  are  given,  or  the  same,  then  the  spaces  arc  inversely 
as  the  quantities  of  matter  ; therefore  x : 92-435  — x : : 
2403-31  — Sx  : 998298  + 8x,  or  by  composition,*  : 92435  : : 
2403-31  — 8*  : 1000701 -31 , and  by  div.  * : 1 ::  2403-3 1 — 8x  : 
10326,  theref,  10826-r =2403-31—  8*,  or  10834*  = 2403-31, 
and  hence  x = -2218  inch  = § of  an  inch  nearly,  or  the  re- 
coil of  the  gun  is  less  than  a quarter  of  an  inch. 

Hence  it  may  be  concluded,  that  so  small  a recoil,  straight 
backwards,  can  have  no  effect  in  causing  the  ball  to  deviate 
from  the  pointed  line  of  direction  : and  that  it  is  very  pro- 
bable we  are  to  seek  for  the  cause  of  this  effect  in  the  bail 
striking  or  rubbing  against  the  sides  of  the  bore,  in  its  passage 
through  it  especially  near  the  exit  at  the  muzzle  ; by  which 
it  must  happen,  that  if  the  ball  strike  against  the  right  side, 
the  ball  will  deviate  to  the  left ; if  it  strike  on  the  left  side, 
it  must  deviate  to  the  right  ; if  it  strike  against  the  under 
side,  it  must  throw  the  ball  upwards,  and  make  it  to  range 
farther  ; but  if  it  strike  against  the  upper  side  it  must  neat 
the  ball  downwards,  and  cause  a shorter  range  : all  which 
irregularities  are  found  to  take  place,  especially  in  guos  that 
have  much  windage,  or  which  have  the  balls  too  small  for 
the  bore. 


PROBLEM  11. 


A ball  of  lead,  of  4 inches  diameter , is  dropped  from  the  top 
of  a tower,  of  65  yards  high,  and  falls  into  a cistern  full  of  wa- 
ter at  the  bottom  of  the  tower,  of  20-  yards  deep  : it  is  required 
to  determine  the  times  of  falling,  both  to  the  surface  and  to  the 
bottom  of  the  water. 

The  fall  in  air  is  195  feet,  and  in  water  60£  feet.  By  the 
common  rules  of  descent,  as  y/  16  : 195  : : I : \ 195  = 

3;  49 


49S 


PROMISCUOUS  EXERCISES. 


3-49  seconds,  the  time  of  descending  in  air.  And  as  16  . 
^ / 196  : : 32  : 8 y'.  196  = 111-71  feet,  the  velocity  at  the  end 
ef  that  time,  or  with  which  the  ball  enters  the  water. 

Again,  by  prob.  22  of  this  vol.  art.  2,  the  space  s Xhyp. 
log.  of  j — ~2,  or  rather  X hyp.  log.  of  — — ^ (the  velocity 
being  decreasing  and  e2  greater  than  a)  X com.  log.  of 


— — - where  n = 11325  the  density  of  lead  n = 1000  that 

“■  a c\  ~ c if  \ 

c . , • 230f/(N /)  , 

oi  water,  a — h — 


3* 


b = — e = 111-71  the  velocity 

’ sa  n ’ 


at  entering  the  water,  and  u the  velocity  at  any  time  after- 
wards, also  of  the  diameter  of  the  ball  = 4 inches,  and  m = 
52  302585  the  hyp.  log.  of  10. 

Hence  then  n = 11325,  n = 1000,  n ' — n — 10325,  d — 


— = - ; then  a = 
12  3 ’ 

^ 3n 9n 9000 


_ 256i/(n  — n)  256  . 10325 


Hdn  8s  90600  151 


Si  .9000 

15  1 

= tit  = To  nearly- 


2931,  and 


Also  e = lll-71; 


therefore  s — 60f  = ~ X log.  of e ° = 5 m X log.  f -. 

zb  v- — a v 3 — a 

This  theorem  will  give  s when  v is  given,  and  by  reverting 
it  will  give  v in  terms  of  s in  the  following  manner. 

• • S — Q 

Dividing  by  5 m,  gives  — = log.  of  —q  = ns,  by  putting 

X C ~ •“ " CL 

n = — ; therefore,  the  natural  number  is  10”f,  = ; 

5m  v2  — a 

■ — d • f 2 _ c[ 

hence  v2  — a = , and?;  = »/  (a  H ),  which,  by  sub- 

l0m  -V  \ ■ jjjni  / j 


stituting  the  numbers  above  mentioned  for  the  letters,  gives 
v = 171 34  for  the  last  velocity,  w'hen  the  space  s = 60|,  or 
when  the  ball  arrives  at  the  bottom  of  the  water. 

But  now  to  find  the  time  of  passing  through  the  water, 
putting  t — aDy  time  in  motion,  and  s and  v the  correspond- 
ing space  and  velocity  the  general  theorem  for  variable  forces 

i .1 

gives  t — — • But  the  above  general  value  of  s being  — X 


hyp.  log. 


or  5 


X hyp.  log. 


e 3 —g 
•v2  — a’ 


therefore  its  fluxioa 


• — 10vv  • s — ■ I0x>  .a  . r 

s = , conseq.  ; fir — = — — , the  correct  fluent  of 

*2  — a v v a 

which  is  — X hvp.  log.  X s/a)  = t the  time, 

which  when  v — 17-134,  or  s = 60f  gives  2-6542  seconds, 
for  the  time  of  descent  through  the  water.  PROBLEM 


PROMISCUOUS  EXERCISES. 


493 


PROBLEM  12. 

Required  to  determine  what  must  be  the  diameter  of  a wa- 
ter-wheel,  so  .as  to  receive  the  greatest  effect  from  a stream  of  wa- 
ter  of  12  feet  fall  ? 

In  the  case  of  an  undershot  wheel 
put  the  height  of  the  water  ab  = 12 
feet  = a,  and  the  radius  bc  or  cd  of 
the  wheel  = x,  the  water'falling  perpen- 
dicularly on  the  extremity  of  the  radius 
cd  at  n.  Then  ac  or  AD=a  — .r,  and  the 
velocity  due  to  this  height,  or  with  which 
the  water  strikes  the  wheel  at  d,  will  be 
asv/(a  — c),  and  the  effect  on  the  wheel  being  as  the  velocity 
and  as  the  length  of  the  lever  cd,  will  be  denoted  bv 
x^/(a — x)  or  y/[ ax2  — x3),  which  therefore  must  be  a maxi- 
mum, or  its  square  ax2  — x3  a maximum.  In  fluxions 
‘iax'x  — 3i2x  = 0 ; and  hence  x = %a  = 8 feet,  the  radius. 

But  if  the  water  be  considered  as 
conducted  so  as  to  strike  on  the  bottom 
of  the  wheel,  as  in  the  annexed  figure, 
it  will  then  strike  the  wheel  with  its 
greatest  velocity,  and  there  can  be  no 
limit  to  the  size  of  the  wheel,  since  the 
greater  the  radius  or  lever  bc,  the 
greater  will  be  the  effect. 

In  the  case  of  an  overshot  wheel, 
a - 2x  will  be  the  fall  of  water,  ^/(a  — 2x) 
as  the  velocity,  and  x (a  — 2x)  or 
^/(ax2  -2x3)the  effect,  then  ax2— 2x3 
is  a maximum,  and  2axx  - 6x2^  = 0 
hence  x — ~a  = 4 feet  is  the  radius  of 
the  wheel. 

But  all  these  calculations  are  to  be  considered  as  independ- 
ent of  the  resistance  of  the  wheel,  and  of  the  weight  of  the 
water  in  the  buckets  of  it. 

PROBLEM  13. 

What  angle  must  a projectile  make  with  the  plane,  of  the 
horizon,  discharged  with,  a given  velocity  v,  so  as  to  describe 
in  its  flight  a parabola  including  the  greatest  area  possible  ? 

By  the  set  of  theorems  (in  art.  92  Projectiles)  for  any 
proposed  angle,  there  can  be  assigned  expressions  for  the 
horizontal  range  and  the  greatest  height  the  projectile  rises 
to,  that  is  the  base  and  axis  of  the  parabolic  trajectory.  Thus, 
putting  s and  c for  the  sine  and  cosine  of  the  angle  of  eleva- 
tion ; 


I 


494  PROMISCUOUS  EXERCISES. 

lion  ; then,  by  the  first  line  of  those  theorems,  the  velocity 
being  v,  the  horizontal  radge  r is  — f-^scv2  ; and,  by  the  4th 
or  last  line  of  theorems*  the  greatest  height  h is  = s2u2. 

But,  by  the  parabola,  § of  the  product  of  the  base  or  range 
and  the  height  Is  the  area,  which  is  now  required  to  be  the 
greatest  possible.  Therefore  .r  X h = xVct2  x s2v 2 must 

be  a maximum,  or,  rejecting  the  constant  factors,  s3c  a maxi- 
mum. But  the  cosine  c,  of  the  angle  whose  sine  is  s,  is 
^/(l— s2)  ; therefore  s3c=s3  ^(1 -,i2)  = ^/ (s6  — s8)  is  the 
maximum,  or  its  square  s6— s6  a maximum.  In  fluxions 
6s5i  — 8s7«  = 0 = 3 — 4s2  ; hence  4s2  = 3,  or  s2  = f , and 
s ==  \ 3 — '8660254,  the  sine  of  60°,  which  is  the  angle  of 

elevation  to  produce  a parabolic  trajectory  of  the  greatesf 
area. 

PROBLEM  14. 

Suppose  a cannon  were  discharged  at  a point  a ; it  is  re- 
quired to  determine  how  high  in  the  air  the  point  c must  be 
raised  above  the  horizontal  line  ab,  so  that  a person  at  c 
letting  fall  a leaden  bullet,  at  the  moment  of  the  cannon's  ex- 
plosion, it  may  arrive  at  b at  the  same  instant  as  he  hear:  the 
report  of  the  cannon,  but  not  till  J ^th  of  a second  after  the  sound 
arrives  at  b : supposing  the  velocity  of  sound  to  be  1140  feet  per 
second,  and  that  the  bullet  falls  freely  without  any  resistance  from 
the  air  ? 

Let  x denote  the  time  in  which  the 
sound  passes  to  c ; then  wiil  x — be 
the  time  in  passing  to  b,  and  x the  time 
also  the  bullet  is  falling  through  cb. 

Then,  by  uniform  motion,  ll40.r  = ac, 
and  1 1 40x  — 1 1 4 = ab  , also  by  descents 
of  gravity,  l3  : x2  : : 16  : 16x2  = bc.  Then,  by  right-angled 
triangles,  ac*  — bc2=ab2  , that  is  1 1402x2  — 162x4  = 1 1402x2 

— 224  X IHOx  + 1 14r , hence.  224  X 1140x—  162x->  = 
1142,  or  1015-3a:  - x4  = 50-77,  the  root  of  which  equa.  is 
x — 10-03  seconds,  or  nearly  10  seconds  ; conseq.  bc  = 16x2 

— 1610  feet  nearly,  the  height  required. 

PROBLEM  15. 

Required  the  quantity,  in  cubic  feet,  of  light  earth,  necessary 
to  form  a bank  on  the  side  of  a canal,  which  witl  just  support 
a pressure  of  water  5 feet  deep,  and  300  feet  long.  And  what 
will  the  carriage  of  the  earth  cost,  at  the  rate  of  1 shilling  per 
ion  ? 


This 


PROMISCUOUS  EXERCISES. 


495 


This  question  may  be  considered  as 
; relating  either  to  water  sustained  by  a 
solid  wail,  or  by  a bank  of  loose  earth. 

In  the  former  case,  let  abc  denote  the 
wall,  sustaining  the  pressure  of  the  water 
behind  it.  Put  the  whole  altitude  ab 
= n,  the  base  bc  or  thickness  at  bottom 
= 6,  any  variable  depth  ad  = \r,  and 
the  thickness  there  de  = y.  Now  the  effect  which  any  num- 
ber of  particles  of  the  fluid  pressing  at  d have  to  break  the 
wall  at  b,  or  to  overturn  it  there,  is  as  the  number  of  particles 
ad  or  x,  and  as  the  lever  bd  = a — x : therefore  the  fluxion 
of  the  effect  of  all  the  forces  is  (a—x)x x — axx—xa  x,  the 
fluent  of  which  is  \ax2  — iar3,  which,  when  x — a,  is  ±a3  for 
the  whole  effect  to  break  or  overturn  the  wall  at  b ; and  the 
effects  of  the  pressure  to  break  at  b and  d.  will  be  as  ab3  and 
ad3.  But  the  strength  of  the  wall  at  d,  to  resist  the  fracture 
there,  like  the  lateral  strength  of  timber,  is  as  the  square  of 
the  thickness,  de2.  Hence  the  curve  line  aec,.  bounding 
the  back  of  the  wall,  so  as  to  be  every  where  equally  strong, 
is  'of  such  a nature,  that  x3  is  always  proportional  to  y- , or  y 

J3 

as  x2 , and  is  therefore  what  is  called  the  semicubical  parabola. 

Now,  to  find  the  area  abc,  or  content  of  the  wall  bounded 
by  this  convex  curve,  the  general  fluxion  of  all  are  as  yx  be- 

3-  . 5.  3. 

comes  x‘-x,  the  fluent  of  which  is  %x 2 = fxx2  = f xy,  that 
is  | of  the  rectangle  ab  X bc  ; and  is  therefore  less  than  the 
triangle  abc,  of  the  same  base  and  height,  in  the  proportion 
of  § to  J-,  or  of  4 to  5. 

But  in  the  case  of  a bank  of  made 
earth,  it  would  not  stand  with  that 
concave  form  of  outside,  if  it  were  ne- 
cessary, but  would  dispose  itself  in  a 
straight  line  ac,  forming  a triangular 
bank  abc.  And  even  if  this  were  not 
the  case  naturally,  it  would  be  proper 
to  make  it  such  by  art  ; because  now 
neither  is  the  bank  to  be  broken  as  with 
lever,  or  overturned  about  the  pivot  or  point  c,  nor  does  it 
resist  the  fracture  by  the  effect  of  a lever,  as  before  ; but,  on 
■ the  contrary,  every  point  is  attempted  to  be  pushed  horizon- 
tally outwards,  by  the  horizontal  pressure  of  the  water,  and 
it  is  resisted  by  the  weight  or  resistance  of  the  earth  at  any 
part  de.  Here  then,  by  hydrostatics,  the  pressure  of  the 
water  against  any  point  d,  is  as  the  depth  ad  ; and,  in  the 
triangle  of  earth  ade,  the  resisting  quantity  in  de  is  as  de, 

; which 


the  effect  of  the 


496 


PROMISCUOUS  EXERCISES. 


which  is  also  proportional  to  ad  by  similar  triangles.  S# 
that,  at  every  point  d in  the  depth,  the  pressure  of  the  water 
and  the  resistance  of  the  soil,  by  means  of  this  triangular 
form  increase  in  the  same  proportion,  and  the  water  and  the-, 
earth  will  every  where  mutually  balance  each  other,  if  at  any 
one  point,  as  b,  the  thickness  bc  of  earth  be  taken  such  as  to 
balance  the  pressure  of  the  water  at  b,  and  then  the  straight 
line  ac  be  drawn,  to  determine  the  outer  shape  of  the  earth. 
All  the  earth  that  is  afterwards  placed  against  the  side  ac,  for 
a convenient  breadth  at  top  for  a walking  path,  &c.  will  also 
give  the  whole  a sufficient  security. 

But  now  to  adapt  these  principles  to  the  numeral  calcula- 
tion proposed  in  the  question  ; the  pressure  of  water  against 
the  point  b being  denoted  by  the  side  ab  = 5 feet,  and  the 
weight  of  water  being  to  earth  as  1000  to  1984,  therefore  as 
1984  : 1000  : : 5 : 2 52  = bc,  the  thickness  of  earth  which 
will  just  balance  the  pressure  of  the  water  there  ; hence  the 
area  of  the  triangle  abc  ~ ^ab  X bc  = X 2-52  = 6-3  ; 
this  mult  by  the  length  300,  gives  1890  cubic  feet  for  the 
quantity  of  earth  in  the  bank  ; and  this  multiplied  by  1984 
ounces,  the  weight  of  1 cubic  foot,  gives,  for  the  weight  of  it, 
3749760  ounces  = 2343601bs  = 104-625  tons  ; the  expense 
of  which,  at  1 shilling  the  ton,  is  5 L 4s.  7 ±d. 

PROBLEM  16. 

A person  standing  at  the  distance  of  20  feet  from  the  bottom 
of  a wall,  which  is  supposed  perfectly  smooth  and  hard,  desires 
to  know  in  what  direction  he  must  throw  an  elastic  ball  against 
it,  with  a velocity  of  80  feet  per  second,  so  that , after  reflection 
from  the  wall,  it  may  fall  at  the  greatest  distance  'possible  from 
the  bottom  on  the  horizantal  plane,  which  is  feet  below  the 
hand  discharging  the  ball  ? 

In  the  annexed  figure  let  dr 
be  the  wall  against  which  the 
ball  is  thrown,  from  the  point 
a,  in  such  a direction,  that  it 
shall  describe  the  parabolic  curve 
ae  before  striking  the  wall,  and 
afterwards  be  so  reflected  as  to  describe  the  curve  ep.  Now 
if  es  be  the  tangent  at  the  point  e,  to  the  curve  ae  describ- 
ed before  the  reflection,  and  ef  the  tangent  at  the  same 
point  to  the  curve  which  the  ball  will  describe  after  re- 
flection, then  will  the  angle  ref  be  = ces  ; and  if  the  curve 
fe  be  produced,  so  as  to  have  gf  for  its  tangent,  it  will  meet 
ac  produced!  in  e,  making  bc  = ac,  and  the  curve  ae  wiil  be 

similar 


PROMISCUOUS  EXERCISES. 


497 


similar  and  equal  to  the  portion  be  of  the  parabola  bep,  but 
turned  the  contrary  way.  Conceiving  either  the  two  curves 
ae  and  f.p,  or  the  continued  curve  bep,  to  be  described  by 
a projectile  in  its  motion,  it  is  manifest  that,  whether  the 
greater  portion  of  the  curve  he  described  before  or  after  the 
ball  reaches  the  wall  dr,  will  depend  on  its  initial  velocity, 
and  on  the  distance  ac  or  bc,  and  on  the  angle  of  projection. 
The  problem  then  is  now  reduced  to  this,  viz.  To  lind  the 
angle  at  which  a ball  shall  be  projected  from  b,  with  a given 
impetus,  so  that  the  distance  dp,  at  which  it  falls,  from  the 
given  point  d on  the  plane  dp,  parallel  to  the  horizon,  shall 
be  a maximum. 

Now  this  problem  may  be 
constructed  in  the  following 
manner  : From  any  point  e 
in  the  horizontal  line  dc,  let 
fall  the  indefinite  perp.  eg,  on  _ 
which  set  off  eb  ==  the  impe-  A 
' tus  corresponding  to  the  given 

velocity,  and  ei  = 2 i the  distance  of  the  horizontal  plane 
below  the  point  of  projection  ; also,  through  i draw  ap  paral- 
lel to  dc.  From  the  point  b set  off  bp  = be  -f-  ei,  and 
bisect  the  angle  ebp  by  the  line  bh’:  then  will  bh  be  the  re- 
quired direction  of  the  ball,  and  if  the  maximum  range  on  the 
plane  ap. 

For,  since  the  ball  moves  from  the  point  b with  the  velo- 
city acquired  by  falling  through  eb,  it  is  manifest,  from  p.136 
this  vol.  that  dc  is  the  directrix  of  the  parabola  described  by 
the  ball.  And  since  both  b and  p are  points  in  the  curve, 
each  of  them  must,  from  the  nature  of  the  parabola,  be  as  far 
from  the  forces  as  it  is  from  the  directrix  ; therefore  b and  p 
will  be  the  greatest  distance  from  each  other  when  the  focus 
f is  directly  between  them,  that  is,  when  ep  = be  -f-  cp. 
And  when  bp  is  a maximum,  since  ei  is  constant,  it  is  obvious 
that  ip  is  a maximum  too.  Also,  the  angle  fbh  being  = ebh, 
the  line  bh  is  a tangent  to  the  parabola  at  the  point  b,  and 
consequently  it  is  the  direction  necessary  to  give  the  range  ip. 

Cor.  1.  When  b coincides  with  i,  ip  will  be  = bp  = be 
+ ei  — 2ei,  and  the  angle  ebh  will  be  45°  : as  is  also  mani- 
fest from  the  common  modes  of  investigation. 

Cor.  2.  When  the  impetus  corresponding  to  the  initial  ve- 
locity of  the  ball  is  very  great  compared  with  ac  or  bc  (fig.  1), 
then  the  part  ae  of  the  curve  will  very  nearly  coincide  with 
jj  its  tangent,  and  the  direction  and  velocity  at  a may  be  account- 
ed the  same  as  those  at  e without  any  sensible  error.  In  this 

Vol.  II.  61 


case 


498 


PROMISCUOUS  EXERCISES. 


case  too  the  impetus  be  (tig.  2)  will  be  very  great  compared 
with  bi,  and  consequently,  e and  i nearly  coinciding,  the  an- 
gle ebh  will  differ  but  little  from  45°. 

Calcul.  From  the  foregoing  construction  the  calculation 
will  be  very  easy.  Thus,  the  first  velocity  being  80  feet  = v. 

Or,  v/  gQ 

then  (art.  92  Projectiles)  — — ~ 99'48186  = EE  the 

impetus  ; hence  ei  = fp  = 101  98186,  and  bp  = be  + ei  = 
201-46372.  Now,  in  the  right-angled  triangle  bip,  the  sides 
bi  and  bp  are  known,  hence  ip  = 201-4482,  and  the  angle 
ibp  = 89°  17'  20"  : half  the  suppl.  of  this  angle  is  45°21  20’' 
= ebh.  And,  in  fig.  1,  ip  — id  = 201-4482  — 10  = 
191-4482  — dp  ; the  distance  the  ball  falls  from  the  wall  after 
reflection. 

PROBLEM  17. 

From  what  height  above  the  given  point  a must  an  elastic  ball 
be  suffered  to  descend  freely  by  gravity,  so  that,  after  striking 
the  hard  plane  at  b,  it  may  be  reflected  back  again  to  the  point 
a,  in  the  least  time  possible  from  the  instant  of  dropping  it  ? 


Let  c be  the  poiDt  required  ; and  put  ac  = x,  and 
ab  = a ; then  i a/  CB  = i v/  (<*+*)  »s  the  time  in  cb, 
and  i y/  ca  = l x is  the  time  in  ca  ; therefore 
i (a+a)  — j y/  x is  the  time  down  ab,  or  the  time 
of  rising  from  b to  a again  : hence  the  whole  time  of 
falling  through  cb  and  returning  to  a,  is  i ^ (a+ar) 
— a y/  x,  which  must  be  a min.  or  2 (a+a;)  - y/x 

a minimum,  in  fluxions  — , — — — 0,  and 

x = \a,  that  is,  ac  = ^ab. 


C 

A 


E 

hence 


PROBLEM  18. 


Given  the  height  of  an  inclined  plane  ; required  its  length,  so 
that  a given  power  acting  on  a given  weight,  in  a direction,  pa- 
rallel to  the  plane,  may  draw  it  up  in  the  least  time  possible. 


Let  a denote  the  height  of  the  plane,  x its  length,  p the 
power,  and  w the  weight.  -Now  the  tendency  down  the  plane 

p 

is  = — , hence  p — — = the  motive  force,  and x — 

x x p-fw 

jX  — aui  __  acceierating  force/ ; hence,  by  the  theorems  for 

(p  rW)X 

constant  forces  (See  lntroduc.  to  Prac.  Ex.  on  Forces)*2  =-/.= 


PROMISCUOUS  EXERCISES. 


499 


<-/’+w):r  mus^  ^ a minimum  or  JS-—  a min.  ; in  fluxions, 
{px—anu)g  px—aw 

2 (px  — aw)xx — px2x  ~ 0,  or  px  ==  2a®,  and  hence p : 
w ::  2 a : x \ : double  the  height  of  the  plane  to  its  length. 

PROBLEM  19. 


■ J1  cylinder  of  oak  is  depressed  in  water  till  its  top  is  just 
-Level  with  the  surface,  and  then  is  suffered  to  ascend  ; it  is 
required  to  determine  the  greatest  altitude  to  which  it  will  rise, 
and  the  time  of  its  ascent. 

Let  a = the  length,  and  b the  area  or  base  of  the  cylinder, 
m the  specific  gravity  of  oak,  that  of  water  being  1,  also  x 
any  variable  height  through  which  the  cylinder  has  ascended. 
Then,  a — x being  the  part  still  immersed  in  the  water, 
( a — x ) X b X 1 = (a  — x)  b is  the  force  of  the  water  upwards 
to  raise  the  cylinder  ; and  a X 6 X m — abm  is  the  weight 
of  the  cylinder  opposing  its  ascent  ; therefore  the  efficacious 
I force  to  raise  the  cylinder  is  (a  — x')  b — abm  ; and,  the  mass 
being  aim,  the  accelerating  force  is 


(a  — x)b — obrr, a — z — an  — x 

abm  g-.,i  c 


=/. 


putting  n — 1 — m the  difference  between  the  specific  gra- 
vities of  water  and  oak. 

Now  if  v denote  the  velocity  of  ascent  at  the  same  time 
when  x space  is  ascended,  then  by  the  theorems  for  variable 


forces,  vv  — 32 ff 


32 

— X (a/ix  — xx),  therefore 
am  - J 


v-  = — X (2  anx  — x2),  and  5 = 87  — : but  when 
am  2am 

the  cylinder  has  acquired  its  greatest  ascent,  v and  v3  = 0, 
therefore  2 anx  — x 2 — 0,  and  hence  x = 2 an  the  part  of  the 
cylinder  that  rises  out  of  the  water,  being  — -15a  or  of 
its  length. 

To  find  when  the  velocity  is  the  greatest,  the  factor  2 anx 
— x3  in  the  velocity  must  be  a max.  then  2 anx  — %xx  — 0, 
and  x = an,  being  the  height  above  the  water  when  the  ve- 
locity is  the  greatest,  and  which  it  appears  is  just  equal  to  the 
half  of  2 an  above  found  for  the  greatest  rise,  when  the  up 
ward  motion  ceases,  and  the  cylinder  descends  again  to  the 
same  depth  as  at  first,  after  which  it  again  returns  ascending 
as  before  ; and  so  on,  continually  playing  up  and  down  to  the 
same  highest  and  lowest  points,  like  the  vibrations  of  a pen- 
dulum, the  motion  ceasing  in  both  cases  in  a similar  man- 
ner at  the  extreme  points,  then  returning,  it  gradually  acce- 
lerates till  arriving  at  the  middle  point,  where  it  is  the 
greatest,  then  gradually  retarding  all  the  way  to  the  next 

extremity 


500 


PROMISCUOUS  EXERCISES. 


extremity  of  the  vibration,  thus  making  all  the  vibrations  in 
equal  times,  to  the  same  extent  between  the  highest  and 
lowest  points,  except  that,  by  the  small  tenacity  and  friction 
&c  of  the  water  against  the  sides  of  the  cylinder,  it  will  be 
gradually  and  slowly  retarded  in  its  motion,  and  the  extent 
of  the  vibrations  decrease  till  at  length  the  cylinder,  like  the 
pendulum,  come  to  rest  in  the  middle  point  of  its  vibrations, 
where  it  naturally  floats  in  its  quiescent  state,  with  the  part 
na  of  its  length  above  the  water. 

The  quantity  of  the  greatest  velocity  will  be  found,  by 
substituting  na  for  x,  in  the  general  value  of  the  velocity 
2 anX  — x2  . a 

v-  ^ when  it  becomes  871^/  — =^^/  a very  nearly, 

the  value  of  m being  -925,  and  consequently  that  of  n — 1 — 
m = -075. 


To  find  the  time  t answering  to  any  space  x.  Here 

-rr,and  by  the  13th 


x 

t = - 

V 


2 nax—x* 
b */  — — - 
2 ma 


__ 

*2  V <M 


32  " v'  (2 nax-x2)’ 
form  the  fluent  is  t =;  | v/2ma  X a,  where  a denotes  the 


circular  arc  to  radius  1 and  versed  sine  — 


Now  at  the  mid' 


71  a 


die  of  a vibration  x is  = na,  and  then  the  vers.  — = — = I 

na  na 


the  radius,  and  a is  the  quadrantal  arc  = 1-5708  ; then  the 
flu.  becomes  } 2ma  X 1 5708  = -17  a X 1-5708=  267^/a 
for  the  time  of  a semivibration  ; hence  the  time  of  each  whole 
vibration  is  -534  a = T8T  a,  which  time  therefore  depends 
on  the  length  of  the  cylinder  a.  To  make  this  time  = 1 
second,  a must  be  = (V5)2  very  nearly  = 34  feet  or  42  inches. 
That  is,  the  oaken  cylinder  of  42  inches  length  makes  its 
vertical  vibrations  each  in  1 second  of  time,  or  is  isochronous 
with  a common  pendulum  of  S9i  inches  long,  the  extent  of 
each  vibration  of  the  former  being  6T3^  inches. 


PROBLEM  20. 

Required  io  determine  the  quantity  of  matter  in  a sphere,  tht 
density  varying  as  the  nth  power  of  the  distance  from  the  centre  2 

Let  r denote  the  radius  of  the  sphere,  d the  density  at 
its  surface,  a = 3-1416  the  area  of  a circle  whose  radius  is  1, 
and  x any  distance  from  the  centre.  Then  4ax2  will  be  the 
surface  of  a sphere  whose  radius  is  x which  may  be  consL 
dered  by  expansion  as  generating  the  magnitude  of  the  solid  . 
therefore  4ax2z  will  be  the  fluxion  of  the  magnitude  ; but 

as 


PROMISCUOUS  EXERCISES. 


501 


as  r" 


dx11 

xn  : : d : — the  density  at  the  distance  x,  therefore 

rn 


= the  fluxion  of  the  mass,  the 

4 adr3 


. „ • ,,  dx n 4 aux"+21- 

4ax2x  X — = 

rn  rn 

% 4 ciclx^^  _ruur  _ 

fluent  of  which — — , when  x = r,  is  — , the  quan- 

(n-f3)r»’  ’ ii  + 3 H 

tity  of  the  matter  in  the  whole  sphere. 

Carol.  1.  The  magnitude  of  a sphere  whose  radius  is  r 
being  far3,  which  call  m ; then  the  mass  or  solid  content  will 

be  X m,  and  the  mean  density  is  — . 

71+3  ’ f 71+3 

Carol.  2.  It  having  been  computed,  from  actual  experi- 
ments, that  the  medium  density  of  the  whole  mass  of  the 
earth  is  about  twice  the  density  d at  the  surface,  we  can 
now  determine  what  is  the  exponent  of  the  decreasing  ratio 
of  the  density  from  the  centre  to  the  circumference,  sup- 
posing it  to  decrease  by  a regular  law,  viz.  as  rn  • for  then  it 

3 d 

will  be  2d  = , and  hence  n — — So  that,  in  this 

72  + 3 

case  the  law  of  decrease  is  as  r 3 
ly  as  the  f ths  power  of  the  radius. 


, or  as  — , that  is,  inverse- 


PROBLEM  21. 

Required  to  determine  where  a body  moving  down  the  convex 
side  of  a cycloid , will  fly  off  and  quit  the  curve. 

Let  aveb  represent  the  cy- 
cloid, the  properties  of  which 
may  be  seen  at  arts.  146  and 
147  this  vol.  and  vdc  its  genera- 
ting semicircle.  Let  e be  the 
point  where  the  motion  com- 
mences, whence  it  moves  along  the  curve,  its  velocity  in- 
creasing both  on  the  curve,  and  also  in  the  horizontal  direc- 
tion df,  till  it  come  to  such  a point,  f suppose,  that  the 
velocity  in  the  latter  direction  is  become  a constant  quantity, 
then  that  will  be  the  point  where  it  will  quit  the  cycloid,  and 
afterwards  describe  a parabola  fg,  because  the  horizontal  ve- 
locity in  the  latter  curve  is  always  the  same  constant  quantity, 
(by  art.  76  Projectiles.) 

Put  the  diameter  vc  = d,  vh  — a,  vi=  x ; then  vd  = dx. 
and  id  ~ y'  (dx — a:2).  Now  the  velocity  in  the  curve  at  r- 
in  descending  down  ef,  being  the  same  as  by  falling  through 
hi  or  x — a , by  art,  139,  will  be  = 8 (x  — «j  ; but  this  ve- 


V 


x 

A \tt 

A 

V 

C BG 


502 


PROMISCUOUS  EXERCISER 


locity  in  the  curve  at  f,  ie  to  the  horizontal  velocity  there, 
as  vd  to  id,  because  vd  is  parallel  to  the  curve  or  to  the 
tangent  at  f,  that  is  dx  : ( dx  — x2)  : : 8 (x  — a)  . | 

^ ^ ^ (ftffjSffd which  is  the  horizontal  velocity  at  f,  j 

Cl 

where  the  body  is  supposed  to  have  that  velocity  a constant  jl 
quantity  ; therefore  also  (x  — of  X (d  — x),  as  well  as  « 

(x — a)  X (d—x~)  = ax  -j-  dx — od  — x 2 is  a constant  quan-  ■ 
tity,  and  also  ax  -j-  dx  — x-  : but  the  fluxion  of  a constant  1 
quantity  is  equal  to  nothing,  that  is  ax  -f-  dx  — 2xx  = 0 = 
a + d — 2x,  and  hence  x = 4*  id  = vi,  the  arithmetical  . 

mean  between  vh  and  vc. 

If  the  motion  should  commence  at  v,  then  x or  vi  would  ! 
be  = Id,  and  i would  be  the  centre  of  the  semicircle. 

PROBLEM  22. 

If  a body  begin  to  move  from  a,  with  a given  velocity,  along  j 
the  quadrant  of  a circle  ab  ; it  is  required  to  show  at  what  point  j 
it  will  fly  off  from  the  curve. 

Let  d denote  the  point  where  the 
body  quits  the  circle  adb,  and  then  de- 
scribes the  parabola  be.  Draw  the  or- 
dinate df,  and  let  ga  be  the  height 
producing  the  velocity  at  a.  Put  ga  = a, 
ac  or  cd  — r,  af  = 37  ; then  the  velo- 
city in  the  curve  at  d will  be  the  same 
as  that  acquired  by  falling  through  gf 
or  a + x,  which  is,  as  before,  ^^/(a+x)  ; 
but  the  velocity  in  the  curve  is  to  the  horizontal  velocity  as 
d?i  to  mu  or  as  cd  to  cf  by  similar  triangles,  that  is,  as 

r : r — x : : 8 (x -fa)  : 8 (x+a)  X --  X , which  is  to 

be  a constant  quantity  where  the  body  leaves  the  circle, 
therefore  also  (r  — x)  (x+a)  and  (r  — x)2  X (x-)-a)  a con- 
stant quantity  ; the  fluxion  of  which  made  to  vanish,  gives 
r—2a 

X — — — = af. 

5 

Hence,  if  a = 0,  or  the  body  only  commence  motion  at  a, 
then  x — ±r,  or  af  = ^ac  when  it  quits  the  circle  at  d.  But 
if  a or  g a were  — or|  ac,  then  r — 2a  = 0,  and  the  body 
would  instantly  quit  the  circle  at  the  vertex  a,  and  describe 
a parabola  circumscribing  it,  and  having  the  same  vertex  a. 


PROBLEM 


PROMISCUOUS  EXERCISES. 


5.03 


PROBLEM  23. 

To  determine  the  position  of  a har  or  leam  ab,  being  supported 
in  equilibria  by  two  cords  ac,  bc,  having  their  two  ends  fixed  in 
the  beam,  at  a and  b. 

By  art.  210  Statics,  the  position 
will  be  such,  that  its  centre  of  gra- 
vity g will  be  in  the  perpendicular 
or  plumb  line  cg. 

Carol.  1.  Draw  gd  parallel  to  the 
cord  ac.  Then  the  triangle  cgd, 
having  its  three  sides  in  the  directions 
of,  or  parallel  to,  the  three  forces,  viz. 
the  weight  of  the  beam,  and  the  ten- 
sions of  the  two  cords  ac,  bc,  these  three  forces  will  be  pro- 
portional to  the  three  sides  cg,  gd,  cd,  respectively,  by  art. 
44  ; that  is,  cg  is  as  the  weight  of  the  beam,  gd  as  the  ten- 
sion or  force  of  ac,  and  cd  as  the  tension  or  force  of  bc. 

Corol.  2.  If  two  planes  eaf,  hbi,  perpendicular  to  the 
two  cords,  be  substituted  instead  of  these,  the  beam  will  be 
still  supported  by  the  two  planes,  just  the  same  as  before  by 
the  cords  because  the  action  of  the  planes  is  in  the  direction 
perpendicular  to  their  surface  ; and  the  pressure  on  the  planes 
will  be  just  equal  to  the  tension  or  force  of  the  respective 
cords.  So  that  it  is  the  very  same  thing,  whether  the  body 
is  sustained  by  the  two  cords  ac,  bc,  or  by  the  two  planes 
ef,  hi  ; the  directions  and  quantities  of  the  forces  acting  at 
a and  b being  the  same  in  both  cases. — Also,  if  the  body  be 
made  to  vibrate  about  the  point  c,  the  points  a,  b will  de- 
scribe circular  arcs  coinciding  with  the  touching  planes  at  a, 
b ; and  moving  the  body  up.  aud  down  the  planes,  will  be  just 
the  same  thing  as  making  it  vibrate  by  the  cords  ; consequent- 
ly the  body  can  only  rest,  in  either  case,  when  the  centre  of 
gravity  is  in  the  perpendicular  cg. 

PROBLEM  24. 

To  determine  the  position  of  the  beam  ab,  hanging  by  one  cord 
acb,  having  its  ends  fastened  at  a and  b,  and  sliding  freely  over  a 
tack  or  pulley  fixed  at  c. 

g being  the  centre  of  gravity  of  the  beam,  cg  will  be  per- 
pendicular to  the  horizon,  as  in  the  last  problem  Now  as 

the 


504 


PROMISCUOUS  EXERCISES. 


the  cord  a gb  moves  freely  about  the  point 
c,  the  tension  of  the  cord  is  the  same  in 
every  part,  or  the  same  both  in  ac  and  bc. 

Draw  gd  parallel  to  ac  : then  the  sides  of 
the  triangle  cgd  are  proportional  to  the 
three  forces,  the  weight  and  the  tensions 
of  the  string  ; that  is,  cd  and  dg  are  as 
the  forces  or  tensions  in  cb  and  ca.  But 
these  tensions  are  equal  ; therefore  cd  = dg,  and  conseq. 
the  opposite  angles  dcg  and  dgc  are  also  equal  ; but  the  an- 
gle dgc  is  = the  alternate  angle  acg  ; theref.  the  angle  acg 
= bcg  ; and  hence  the  line  cG  bisects  the  vertical  angle  acb. 
and  conseq.  ac  : cb  : : ag  : gb. 

PROBLEM  25. 


To  determine  the  position  of  the  beam  ab,  moveable  about  the 
end  b,  and  sustained  by  a given  weight  g,  hanging  by  a cord  ac g, 
going  over  a pulley  at  c,  and  fixed  to  the  other  end  a. 

Let  w — the  weight  of  the  beam, 
and  g denote  the  place  of  its  cen- 
tre of  gravity.  Produce  the  direc- 
tion of  the  cord  ca  to  meet  the 
horizontal  line  be  in  d ; also  let 
fall  ae  perp.  to  be  : then  ae  is  the 
direotion  of  the  weight  of  the  beam,  and  da  the  direction  of 
the  weight  g,  the  former  acting  at  g by  the  lever  bg,  and 
the  latter  at  a by  the  lever  ba  ; theref.  the  intensity  of  the 
former  is  tjd  X bg  and  that  of  the  latter  g X ba;  but  these 
are  also  proportional  to  the  sines  of  their  angles  of  direction 
with  ab,  that  is,  of  the  angles  bae,  and  bad  ; therefore  the 
whole  intensity  of  the  former  is  w X bg  X sin.  bae,  and  of 
the  latter  it  is  g X ba  X sin.  bad.  But,  since  these  two 
forces  balance  each  other,  they  are  equal,  viz  w X bg  X sin. 
bae  = g X ba  X sin.  bad,  and  therefore  w : g : : ba  X sin. 
bad  : bg  X sin.  bae,  or  w X bg  : g X ba  : : sin.  bad  : 
sin  bae. 


PROBLEM  26. 

To  determine  the  position  of  the  beam  ab,  sustained  by  the  given 
weights  m,  n,  by  means  of  the  cords  ac  m,  bdh,  going  over  the  fix- 
ed pulleys  c,  d.  - 


it 


K 


\D 


G 


Let 


PROMISCUOUS  EXERCISES. 


5Q5 


Let  g be  the  place  of  the  centre 
of  gravity  of  the  beam.  Now  the 
effect  of  the  weight  m,  is  as  m, 
and  as  the  lever  ag,  and  as  the 
sine  of  the  angle  of  direction  a ; 
and  the  effect  of  the  weight  n,  is 
as  n,  and  as  the  lever  eg  and  as 
the  sine  of  the  angle  of  direction  b ; but  these  two  effects 
are  equal,  because  they  balance  each  other  ; that  is,  m X ag 
X sin.  a = n X bg  X sin.  b ; theref.  m X ag  : n X bg  : : 
sin.  b : sin.  a. 

PROBLEM  27. 

To  determine  the  position  of  the  two  posts  ad  and  be,  sup- 
porting the  beam  ab,  so  that  the  beam  may  rest  in  equilibrio. 

Through  the  centre  of  gravity 
! c of  the  beam,  draw  cg  perp.  to 
the  horizon  ; from  any  point  c 
in  which  draw  cad,  cbe  through 
the  extremities  of  the  beam  ; then 
ad  and  be  will  be  the  positions 
of  the  two  posts  or  props  re- 
quired, so  as  ab  may  be  sustained 
in  equilibrio  ; because  the  three 
forces  sustaining  any  body  in  such  a state,  must  be  all  directed 
to  the  same  point  c. 

Corol.  If  gf  be  drawn  parallel  to  cd  ; then  the  quantities 
of  the  three  forces  balancing  the  beam,  will  be  proportional 
to  the  three  sides  of  the  triangle  cgf,  viz  cg  as  the  weight 
of  the  beam,  cf  as  the  thrust  or  pressure  in  be,  and  fg  as  the 
thrust  or  pressure  in  ad. 

Scholium.  The  equilibrium  may  be  equally  maintained  by 
the  two  posts  or  props  ad,  be,  as  by  the  two  cords  ac,  bc, 
or  by  two  planes  at  a and  b perp.  to  those  cords.— -It  does  not 
always  happen  that  the  centre  of  gravity  is  at  the  lowest  place 
to  which  it  can  get,  to  make  an  equilibrium  ; for  here  when 
the  beam  ab  is  supported  by  the  posts  da,  eb,  the  centre  of 
gravity  is  at  the  highest  it  can  get  ; and  being  in  that  posi- 
tion, it  is  not  disposed  to  move  one  way  more  than  another, 
and  therefore  is  as  truly  in  equilibrio,  as  if  the  centre  was  at 
the  lowest  point.  It  is  true  this  is  only  a tottering  equili- 
brium, and  any  the  least  force  will  destroy  it  ; and  then,  if 
the  beam  and  posts  be  moveable  about  the  angles  a,  b,  d,  e, 

Voe.  II.  65  which 


50(3 


PROMISCUOUS  EXERCISES. 


which  is  all  along  supposed,  the  beam  will  descend  till  it  is 
below  the  points  d,  e,  and  gain  such  a position  as  is  described 
in  prob.  26,  supposing  the  cords  fixed  at  c and  d,  in  the  fig. 
to  that  prob.  and  then  g will  be  at  the  lowest  point,  coming 
there  to  an  equilibrium  again.  In  planes,  the  centre  of  gra- 
vity g may  be  either  at  its  highest  or  lowest  point.  And 
there  are  cases,  when  that  centre  is  neither  at  its  highest  nor 
lowest  point,  as  may  happen  in  the  case  of  prob.  24. 

PROBLEM  28. 

Supposing  the  beam  ab  hanging  by  a pin  at  b,  and  lying  on  the 
wall  ac  ; it  is  required  to  determine  the  forces  or  pressures,  at 
the  points  a and  b,  and  their  directions. 

Draw  ad  perp.  to  ab,  and  through 
g,  the  Gentre  of  gravity  of  the  beam, 
draw  gd  perp.  to  the  horizon  ; and 
join  bd  Then  the  weight  of  the 
beam,  and  the  two  forces  or  pres- 
sures at  a and  b,  will  be  in  the  di- 
rections of  the  three  sides  of  the 
triangle  adg  ; or  in  the  directions 
of,  and  proportional  to,  the  three 
sides  of  the  triangle  gdh,  having 
drawn  gh  parallel  to  bd  ; viz.  the  weight  of  the  beam  as  gd, 
the  pressure  at  a as  hd,  and  the  pressure  b as  gh,  and  in 
these  directions 

For,  the  action  of  the  beam  is  in  the  direction  gd  ; and 
the  action  of  the  wall  at  a,  is  in  the  perp  ad  ; conseq.  the 
stress  on  the  pin  at  b must  be  in  the  direction  bd,  because 
all  the  three  forces  sustaining  a body  in  equilibrio,  must  tend 
to  the  same  point,  as  d. 

Carol.  1.  If  the  beam  were  supported  by  a pin  at  a,  and 
laid  upon  the  wall  at  b ; the  like  construction  must  be  made 
at  b.  as  has  been  done  at  a,  and  then  the  forces  and  their 
directions  will  be  obtained. 

Corol  2.  It  is  all  the  same  thing,  whether  the  beam  is 
sustained  by  the  pin  b and  the  wall  ac,  or  by  two  cords  be, 
af,  acting  in  the  directions  bd,  da,  and  with  the  forces 

HG,  HD. 


PROBLEM  29. 

To  determine  the  Quantities  and  Directions  of  the  Forces 
exerted  by  a heavy  beam  ab,  at  its  two  Extremities  and  its 
Centre  of  Gravity,  bearing  against  a perp.  wall  at  its  upper 
end  b. 


Front 


PROMISCUOUS  EXERCISES. 


50'; 


From  b draw  bc  perp.  to  the  face  of 
ihe  wall  be,  which  will  be  the  direction 
of  the  f ree  at  b ; also  through  g,  the 
centre  of  gravity,  draw  cgd  perp.  to  the 
horizontal  line  ae,  then  cd  is  the  direc- 
i tinn  of  the  weight  of  the  beam  ; and  be- 
cause, these  two  forces  meet  in  the  point 
c,  the  third  force  or  push  a,  must  he  in  ca,  directlj  from  c • 
so  that  the  three  forces  are  in  the  directions  cd,  bc,  ca,  or  in 
the  directions  cd.  da,  ca  ; and.  these  last  three  forming  a tri- 
angle, the  three  forces  are  not  only  in  those  directions,  but 
are  also  proportional  to  these  three  lines  ; viz.  the  weight  in 
or  on  the  beam,  as  the  line  cd  ; the  push  against  the  wall  at 
b,  as  the  horizontal  line  ad  ; and  the  thrust  at  the  bottom,  as 
the  line  ac. 

Some  of  the  foregoing  problems  will  he  found  useful  in  dif- 
ferent cases  of  carpentry,  especially  in  adapting  the  framing 
of  the  roofs  of  buildings,  so  as  to  be  nearest  in  equilibrio  in 
all  their  parts.  And  the  last  problem,  in  particular,  will  be 
very  useful  in  determining  the  push  or  thrust  of  any  arch 
against  its  piers  or  abutments,  and  thence  to  assign  their 
thickness  necessary  to  resist  that  push.  The  following  pro- 
blem will  also  be  of  great  use  in  adjusting  the  form  of  a 
mansard  roof,' or  of  an  arch,  and  the  thickness  of  every  part, 
so  as  to  be  truly  balanced  in  a state  of  just  equilibrium. 

PROBLEM  30. 

Let  there  he  any  number  of  lines,  or  bars,  or  beams,  ab,  bc, 
od,  de,  <$'C.  all  in  the  same  vertical  plane,  connected  together 
and  freely  moveable  about  the  joints  or  angles  a,  b,  c,  d,  e,  4'c. 
and  kept  in  equilibrio  by  their  own  weights,  or  by  weights  only 
laid  on  the  angles:  It  is  required  to  assign  the  proportion  of 

those  weights  ; as  also  the  force  or  push  in  the  direction  of  the 
said  lines  ; and  the  horizontal  thrust  at  every  angle. 

Through  any 
point,  as  d,  draw 
a vertical  line 
flDHg,  &c.  : to 

which,  from  any 
point,  as  c,  draw 
lines  in  the  direc- 
tion of,  or  paral- 
lel to,  the  given  lines  or  beams,  viz.  ca  parallel  to  ae,  and  cb 
parallel  to  bc,  and  ce  to  de,  and  cf  to  ef,  and  eg  to  fg,  &lc.  • 

a!sp 


508 


PROMISCUOUS  EXERCISES. 


also  ch  parallel  to  the  horizon,  or  perpendicular  to  the  xc, 
tical  line  aDg,  iD  which  also  all  these  parallels  terminate. 

Then  will  all  those  lines  be  exactly  proportional  to  the 
forces  acting  or  exerted  in  the  directions  to  which  they  are 
parallel,  and  of  ail  the  three  kinds,  viz.  vertical,  horizontal, 
and  oblique.  That  is,  the  oblique  forces  or  thrusts  in  direc- 
tion of  the  bars ab,  bc,  cd,  de,  ef,  fg, 

are  proportional  to  their  parallels  ca,  c b,  cd,  ce,  cf,  eg  ; 

and  the  vertical  weights  on  the  angles  b,  c,  d,  e,  f,  &c. 

are  as  the  parts  of  the  vertical  . . ab,  od,  oe,  ef,  fg, 

and  the  weight  of  the  whole  frame  abcdefg, 
is  proportional  to  the  sum  of  all  the  verticals,  or  to  ag  ; also 
the  horizontal  thrust  at  every  angle,  is  every  where  the  same 
constant  quantity,  and  is  expressed  by  the  constant  horizon- 
tal line  cu. 

Demonstration.  All  these  proportions  of  the  forces  derive 
and  follow  immediately  from  the  general  well-known  pro- 
perty in  Statics,  that  when  any  forces  balance  and  keep  each 
other  in  equilibrio,  they  are  respectively  in  proportion  as  the 
lines  drawn  parallel  to  their  directions,  and  terminating  each 
other. 

Thus,  the  point  or  angle  b is  kept  in  equilibrio  by  three 
forces,  viz.  the  weight  laid  and  acting  vertically  downward 
on  that  point,  and  by  the  two  oblique  forces  or  thrusts  of  the 
two  beams  ab,  cb,  and  in  these  directions.  But  ca  is  parallel 
to  ab,  and  cb,  to  bc,  and  ab,  to  the  vertical  weight  ; these 
three  forces  are  therefore  proportional  to  the  three  lines  ab, 
ca,  cb. 

In  like  manner,  the  angle  c is  kept  in  its  position  by  the 
weight  laid  and  acting  vertically  on  it,  and  by  the  two  oblique 
forces  or  thrust  in  the  direction  of  the  bars  bc,  cd  : conse- 
quently these  three  forces  are  proportional  to  the  three  lines 
J>d,  cb,  cd,  which  are  parallel  to  them. 

Also,  the  three  forces  keeping  the  point  d in  its  position, 
are  proportional  to  their  three  parallel  lines  dc,  cd,  ce.  And 
the  three  forces  balancing  the  angle  e,  are  proportional  to 
their  three  parallel  lines  ef,  ce,  cf.  And  the  three  forces 
balancing  the  angle  f.  are  proportional  to  their  three  parallel 
Vines  fg,  cf,  eg.  And  so  on  continually,  the  oblique  forces 
or  thrusts  in  the  directions  of  the  bars  or  beams,  being  always 
proportional  to  the  parts  of  the  lines  parallel  to  them,  inter- 
cepted by  the  common  vertical  line  ; while  the  vertical  forces 
or  weights,  acting  or  laid  on  the  angles,  are  proportional  to 
the  parts  of  this  vertical  line  intercepted  by  the  two  lines  pa- 
rallel lo  the  lines  61  the  corresponding  angles. 

Again,  with  regard  to  the  horizontal  force  or  thrust  : since 

the 


PROMISCUOUS  EXERCISES. 


50&. 


the  line  dc  represents,  or  is  proportional  to  the  force  in  the 
direction  nc,  arising  from  the  weight  or  pressure  on  the  angle 
d ; and  since  the  oblique  force  dc  is  equivalent  to,  and  re- 
solves into,  the  two  dh,  hc,  and  in  those  directions,  by  the 
resolution  of  forces,  viz  the  vertical  force  dh,  and  the  hori- 
zontal force  hc  ; it  fallows,  that  the  horizontal  force  or  thrust 
at  the  angle  d,  is  proportional  to  the  line  ch  ; aod  the  part  of 
the  vertical  force  or  weight  on  the  angle  d,  fvhicb  produces 
the  oblique  force  dc,  is  proportional  to  the  part  of  the  vertical 
■ line  dh. 

In  like  manner,  the  oblique  force  c b,  acting  at  c,  in  the 
direction  sb,  resolves  into  the  two  bn,  hc  ; therefore  the  hori- 
zontal force  or  thrust  at  the  aDgle  c,  is  expressed  by  the  line 
ch,  the  very  same  as  it  was  before  for  the  angle  d ; and  the 
f vertical  pressure  at  c,  arising  from  the  weights  on  both  d and  c, 

| is  denoted  by  the  vertical  line  bn. 

Also,  the  oblique  force  ac,  acting  at  the  angle  b,  in  the 
direction  ba,  resolves  into  the  two  oh,  hc  ; therefore  again 
the  horizontal  thrust  at  the  angle  b,  is  represented' by  the  line 
ch,  the  very  same  as  it  was  at  the  points  c and  d ; and  the 
vertical  pressure  at  b,  arising  from  the  weights  on  b,  c,  and  d, 
is  expressed  by  the  part  of  the  vertical  line  ch. 

Thus  also,  the  oblique  force  ce,  in  direction  de,  resolves 
into  the  two  ch  ne,  being  the  same  horizontal  force  with  the 
vertical  He  ; and  the  oblique  force  cf,  in  direction  ef,  re- 
solves into  the  two  ch,  h/;  and  (he  oblique  force  eg,  in 
direction  fg,  resolves  into  the  two  ch,  h g : and  the  oblique 
force  eg,  in  direction  fg,  resolves  into  the  two  ch,  h g ; and 
I so  on  continually,  the  horizontal  force  at  every  point  being 
expressed  by  the  same  constant  line  ch  ; and  the  vertical 
pressures  on  the  angles  by  the  parts  of  the  verticals,  viz.  an 
| the  whole  vertical  pressure  at  b,  from  the  weights  on  the 
angles  b,  c,  d : and  bn  the  whole  pressure  on  c from  the 
weights  on  c and  d ; and  dh  the  part  of  the  weight  on  d 
causing  the  oblique  force  dc  ; and  He  the  other  part  of  the 
weight  on  d causing  the  oblique  pressure  de  ; and  h f the 
whole  vertical  pressure  at  e from  the  weights  on  d and  e ; and 
h g the  whole  vertical  pressure  on  f arising  from  the  weights 
laid  on  d,  e,  and  f And  so  on. 

So  that,  on  the  whole,  ch  denotes  the  whole  weight  on  the 
points  from  d to  a ; and  Hg  the  whole  weight  on  the  points  from 
d to  g ; and  ag  the  whole  weigbt  on  all  points  on  both  sides  ; 
while  ab,  bo,  ne,  tf,fg  express  the  several  particular  weights, 
laid  on  the  angles  b,  c,  d,  e,  f. 

Also,  the  horizontal  thrust  is  every  where  the  same  cor.- 
L-tant  quantity,  and  is  denoted  by  the  line  ch. 

Lastly, 


PROMISCUOUS  EXERCISES. 


510 

Lastly,  the  several  oblique  forces  or  thrusts,  in  the  direc- 
tions ab,  eg,  cd,  de,  ef,  fg,  are  expressed  by,  or  are  pro- 
portional to,  their  corresponding  parallel  lines,  ca,  cb,  cd,  ce, 
<=/>  cg- 

Lorol.  1.  It  is  obvious,  aDd  remarkable,  that  the  lengths 
of  the  bars  ab,  bc,  &c.  do  not  effect  or  alter  the  proportions 
of  any  of  these  loads  or  thrusts  ; since  all  the  lines  ca,  cb,  ab, 
&c.  remain  the«ame,  whatever  be  the  lengths  of  ab,  bc,  Lc. 
The  positions  of  the  bars,  and  the  weights  on  the  angles  de- 
pending mutually  on  each  other,  as  well  as  the  horizontal 
and  oblique  thrusts.  Thus,  if  there  be  given  the  position  of 
dc,  and  the  weights  or  loads  laid  on  the  angles  d.  c,  b;  set  these 
on  the  vertical,  dh,  d b,  ba,  thenc&,  cagive  the  directions  or 
positions  of  cb,  ba,  as  well  as  the  quantity  or  proportion  ch  of 
the  constant  horizontal  thrust. 

Corol.  2.  If  ch  be  made  radius  ; then  it  is  evident  that 
iia  is  the  tangent,  and  ca  the  secant  of  the  elevation  of  ca  or 
ab  above  the  horizon  ; also  h b is  the  tangent  and  cb  the  se- 
cant of  the  elevation  of  cb  orcB  ; also  hd  and  cd  the  tangent 
and  secant  of  the  elevation  of  cd  ; also  ne  and  ce  the  tangent 
and  secant  of  the  elevation  of  ce  or  de  ; also  h f and  cf  the 
tangent  and  secant  of  the  eleyation  of  ef  ; and  so  on  ; also 
the  parts  of  the  vertical  ab,  bn,  ef,  fg,  denoting  the  weights 
laid  on  the  several  angles,  are  the  differences  of  the  said  tan- 
gents of  elevations.  Hence  then  in  general, 

1st.  The  oblique  thrusts,  in  the  directions  of  the  bars,  are 
to  one  another,  directly  in  proportion  as  the  secants  of  their 
angles  of  elevation  above  the  horizontal  directions  ; or,  which 
is  the  same  thing,  reciprocally  proportional  to  the  cosines  of 
the  same  elevations,  or  reciprocally  proportional  to  the  sines 
of  the  vertical  angles,  a,  b,  d,  e,  f,  g,  &c.  made  by  the  ver- 
tical line  with  the  several  directions  of  the  bars  ; because  the 
secants  of  any  angles  are  always  reciprocally  in  proportion  as 
their  cosines. 

2.  The  weight  or  load  laid  on  each  angle,  is  directly  pro- 
portional to  the  difference  between  the  tangents  of  the  ele- 
vations above  the  horizon,  of  the  two  lines  which  form  the 
angle. 

3.  The  horizontal  thrust  at  every  angle,  is  the  same  con- 
stant quantity,  and  has  the  same  proportion  to  the  weight  on 
the  top  of  the  uppermost  bar,  as  radius  has  to  the  tangent  of 
the  elevation  of  that  bar.  Or,  as  the  whole  vertical  ag,  is  to 
the  line  ch,  so  is  the  weight  of  the  whole  assemblage  of  bars, 
to  the  horizontal  thrust.  Other  properties  also,  concerning 
the  weights  and  the  thrusts,  might  be  pointed  out,  but  they 
are  less  simple  and  elegant  than  the  above,  and  are  therefore 

omitted  r 


PROMISCUOUS  EXERCISES, 


oil 


smitted ; the  following  only  excepted,  which  are  inserted 
here  on  account  of  their  usefulness. 

Corol.  3.  It  may  hence  be  deduced  also,*  that  the  weight 
©r  pressure  laid  on  any  angle,  is  directly  proportional  to  the 
continual  product  of  the  sine  of  that  angle  and  of  the  secants 
of  the  elevations  of  the  bars  or  lines  which  form  it.  Thus, 
in  the  triangle  be d,  in  which  the  side  bn,  is  proportional  to 
the  weight  laid  on  the  angle  c,  because  the  sides  of  any  tri- 
angle are  to  one  another  as  the  sines  of  their  opposite  angles, 
therefore  as  sin.  d : c b : : sin.  be d : bn  ; that  is,  bn  is  as 

s--  '-^£D  X c b ; but  the  sine  of  angle  d is  the  cosine  of  the 
sin  . d 

elevation  dch,  and  the  cosine  of  any  angle  is  reciprocally 
proportional  to  the  secant,  therefore  bn  is  as  sin.  ben  X sec. 
®ch  X cb  ; and  c b being  as  the  secant  of  the  angle  b ch  of 
the  elevation  of  be  or  bc  above  the  horizon,  therefore  bn  is 
as  sin.  ben  X sec.  bcH  X sec.  dch  ; and  the  sine  of  ben 
being  the  same  as  the  sine  of  its  supplement  bcd  ; therefore 
the  weight  on  the  angle  c,  which  is  as  bn,  is  as  the  sin  bcd 
X sec.  dch  X sec.  ben,  that  is,  as  the  continual  product  of 
the  sine  of  that  angle,  and  the  secants  of  the  elevations  of  its 
two  sides  above  the  horizon. 

Corol.  4.  Further,  it  easily  appears  also,  that  the  same 
weight  on  any  angle  c,  is  directly  proportional  to  the  sine  of 
that  angle  bcd,  and  inversely  proportional  to  the  sines  of  ' 
the  two  parts  bcp,  dcp,  into  which  the  same  angle  is  divided 
by  the  vertical  line  cp  For  the  secants  of  angles  are  reci- 
procally proportional  to  their  cosines  or  sines  of  their  com- 
plements : but  bcp  = c&H,  is  the  complement  of  the  eleva- 
tion 6ch,  and  dcp  is  the  complement  of  the  elevation  dch  ; 

, therefore  the  secant  of  ben  X secant  of  dcii  is  reciprocally 
: as  the  sin.  bcp  x sin.  dcp  ; also  the  sine  of  ben  is  — the 
sine  of  its  supplement  bcd  ; consequently  the  weight  on  the 
angle  c,  which  is  proportional  to  sin.  6cd  X sec.  ben  X 

sec.  dch.  is  also  proportional  to  , when  the 

1 1 sin.  bcp  X sin.  dcp 

whole  frame  or  series  of  angles  is  balanced,  or  kept  in  equi- 
librio,  by  the  weights  on  the  angles  ; the  same  as  in  the  pre- 
ceding proposition. 

Scholium.  The  foregoing  proposition  is  very  fruitful  in 
its  practical  consequences,  and  contains  the  whole  theory  of 
arches,  which  may  be  deduced  from  the  premises  by  sup- 
posing the  constituting  bars  to  become  very  short,  like  arch 
stones,  so  as  to  form  the  curve  of  an  arch.  It  appears  too, 

: that  the  horizontal  thrust,  which  is  constant  or  uniformly  the 

same 


S12 


PROMISCUOUS  EXERCISES. 


same  throughout,  is  a proper  measuring  unit,  by  means  of 
which  to  estimate  the  other  thrusts  and  pressures,  as  they 
are  all  determinable  from  it  and  the  given  positions  ; and  the 
value  of  it,  as  appears  above,  may  be  easily  computed  from 
the  uppermost  or  vertical  part  alone,  or  from  the  whole  as- 
semblage together,  or  from  any  part  of  the  whole,  counted 
from  the  top  downwards. 

The  solution  of  the  foregoing  proposition  depends  on  this 
consideration,  viz.  that  an  assemblage  of  bars  or  beams,  being 
connected  together  by  joints  at  their  extremities,  and  freely 
moveable  about  them,  may  be  placed  in  such  a vertical  posi- 
tion, as  to  be  exactly  balanced  or  kept  in  equilibrio,  by  their 
mutual  thrusts  and  pressures  at  the  joints  ; and  that  the  effect 
will  be  the  same  if  the  bars  themselves  be  considered  as  with- 
out weight,  and  the  angles  be  pressed  down  by  laying  on 
them  weights,  which  shall  be  equal  to  the  vertical  pressures 
at  the  same  angles,  produced  by  the  bars  in  the  case  when 
they  are  considered  as  endued  with  tbeir  own  natural  weights. 
And  as  we  have  lound  that  the  bars  may  he  of  any  length, 
without  affecting  the  general  properties  and  proportions  of 
the  thrusts  and  pressures,  therefore  by  supposing  them  to 
become  short,  like  arch  stones,  it  is  plain  that  we  shall  then 
have  the  same  principles  and  properties  accommodated  to  a 
real  arch  of  equilibration,  or  one  that  supports  itself  in  a per- 
fect balance.  It  may  be  further  observed  that  the  conclu- 
sions here  derived,  in  this  proposition  and  its  corollaries, 
exactly  agree  with  those  derived  in  a very  different  way,  in 
my  principles  of  bridges,  viz.  in  propositions  1 and  2,  and 
their  corollaries. 

PROBLEM  31. 

If  the  whole  figure  in  the  last  problem  be  inverted , or  turn- 
ed round  the  horizontal  line  ag  as  an  axis,  till  it  be  completely 
reversed , or  in  the  same  vertical  plane  below  the  first  position, 
each  angle  d,  d,  Sfc.  being  in  the  same  plumb  line  ; and  if 
weights  i,  k,  I,  m,  n,  which  are  respectively  equal  to  the  weights 
laid  on  the  angles , b,  c,  n,  e,  f,  of  the  first  figure , be  now  sus- 
pended by  tltreads  from  the  corresponding  angles  b,  c,  d,  e,  f, 
of  the  lower  figure  ; it  is  required  to  show  that  those  weights  keep 
this  figure  in  exact  equilibrio,  the  same  as  the  former  and  all  the 
tensions  or  forces  in  the  latter  ease,  whether  vertical  or  horizontal 
or  oblique,  will  be  exactly  equal  to  the  eorresponding  forces  oj 
weight  or  pressure  or  thrust  in  the  like  directions  of  the  first 
figure. 


This 


PROMISCUOUS  EXERCISES.  513 


j in  both  cases  Thus,  from  the  equality  of  the  corresponding 
weights,  at  the  like  angles,  the  ratios  of  the  weights,  ab , bd, 
dh , he,  &c.  in  the  lower  figure,  are  the  very  same  as  those  ab, 
bn,  dh,  He,  &c.  in  the  upper  figure  ; and  from  the  equality 
of  the  constant  horizontal  forces  ch.  ch,  and  the  similarity 
of  the  positions,  the  corresponding  vertical  lines,  denoting 
the  weights,  are  equal,  namely,  ab  = ab,  bn  — bd,  dh  = dh, 
&c.  The  same  may  be  said  of  the  oblique  lines  also,  ca,  cb, 
&c.  which  being  parallel  to  the  beams  a b,  be,  &c.  will  denote 
1 the  tensions  of  these  in  the  direction  of  their  length,  the 
same  as  the  o blique  thrusts  or  pushes  in  the  upper  figures. 
Thus,  all  the  corresponding  weights  and  actions  and  posi- 
tions, in  the  two  situations,  being  exactly  equal  and  similar, 
changing  only  drawing  and  tension  for  pushing  and  thrusting, 

; the  oalance  and  equilibrium  of  the  upper  figure  is  still  pre- 
served the  same  in  the  hanging  festoon  or  lower  one. 

Scholium.  The  same  figure,  it  is  evident,  will  also  arise, 
if  the  same  weights,  i,  k,  l,  m,  n,  be  suspended  at  like  dis- 
tances, a b,  be,  &c.  on  a thread,  or  cord,  or  chain,  toe.  having 
in  itself  little  or  no  weight.  For  the  equality  of  the  weights, 
and  their  directions  and  distances,  will  put  the  whole  line, 
when  they  come  to  equilibrium,  into  the  same  festoon  shape 
of  figure.  So  that,  whatever  properties  are  inferred  in  the 
corollaries  to  the  foregoing  prob.  will  equally  apply  to  the  fes- 
toon or  lower  figure  hanging  in  equilibrio. 

This  is  a most  useful  principle  in  all  cases  of  equilibriums, 
especially  to  the  mere  practical  mechanist,  and  enables  him, 
in  an  experimental  way  to  resolve  problems,  which  the  best 
mathematicians  have  found  it  no  easy  matter  to  effect  by  mere 
Vol.  IF.  66 


com- 


514 


PROMISCUOUS  EXERCISES. 


computation.  For  thus,  in  a simple  and  easy  way  he  obtains 
the  shape  of  an  equilibrated  arch  or  bridge  ; and  thus  also 
he  readily  obtains  the  positions  of  the  rafters  hn  the  frame  of 
an  equilibrated  curb  or  mansard  roof ; a single  instance  of 
which  may  serve  to  show  the  extent  and  uses  to  which  it  may 
be  applied.  Thus,  if  it  should  be  required  to  make  a curb 


frame  roof  having  a given  width 
ae,  and  consisting  of  four  rafters 
ab,  bc,  cd,  de,  which  shall  either 
be  equal  or  any  given  proportion 
to  each  other.  There  can  be  no 
doubt  but  that  the  best  form  of 
the  roof  will  be  that  which  puts 

all  its  parts  in  equilibrio,  so  that  there  may  be  no  unbalanced 
parts  which  may  require  the  aid  of  ties  or  stays  to  keep  the 
frame  in  its  position.  Here  the  mechanic  has  nothing  to  do 
but  to  take  four  like  but  small  pieces,  that  are  either  equal 
or  in  the  same  given  proportions  as  those  proposed,  and  con- 
nect them  closely  together  at'the  joints  a,  b,  c,  d,  e,  by  pins 
or  strings,,  so  as  to  be  freely  moveable  about  them  ; then 
suspend  this  from  two  pins  a,  e, 
fixed  in  a horizontal  line,  and  the 
chain  of  the  pieces  will  arrange 
itself  in  such  a festoon  or  form, 
abode,  that  all  its  parts  will  come 
to  rest  in  equilibrio.  Then,  by 
inverting  the  figure,  it  will  ex- 
hibit the  form  and  frame  of  a 
curb  roof  afyJe,  which  will  also 
bei.in  equilibrio,  the  thrusts  of  the 
pieces  now  balancing  each  other, 


in  the  same  manner  as  was  done  by  the  mutual  pulls  or  ten- 


sions of  the  hanging  festoon  abode.  By  varying  the  dis- 
tance ae,  of  the  points  of  suspension,  moving  them  nearer 
to,  or  farther  oif,  the  chain  will  take  different  forms  ; then 
the  frame  abcde  may  be  made  similar  to  that  form  which 
has  the  most  pleasing  or  convenient  shape,  found  above  as  a 
model. 

Indeed  this  principle  is  exceeding  fruitful  in  its  practical 
consequences.  It  is  easy  to  perceive  that  it  contains  the 
whole  theory  of  the  construction  of  arches  : for  each  stone  of 
an  arch  may  be  considered  as  one  of  the  rafters  or  beams  in 
the  foregoing  frames,  since  the  whole  is  sustained  by  the  mere 
principle  of  equilibration,  and  the  method,  in  its  application, 
will  afford  some  elegant  and  simple  solutions  of  the  most  diffi- 
cult cases  of  this  important  problem.  + 


PROBLEM 


PROMISCUOUS  EXERCISES. 


515 


PROBLEM  32. 

Of  all  Hollow  Cylinders,  whose  Lengths  and  the  Diame- 
ters of  the  Inner  and  Outer  Circles  continue  the  same,  it  is 
required  to  show  what  will  be  the  Position  of  the  Inner  Circle 
when  the  Cylinder  is  the  Strongest  Laterally. 

Since  the  magnitudes  of  the  two  circles  are  constant,  the 
area  of  the  solid  space  included  between  their  two  circum- 
ferences, will  be  the  same,  whatever  be  the  position  of  the 
inner  circle,  that  is,  there  is  the  same  number  of  fibres  to  be 
broken,  and  in  this  respect  the  strength  will  be  always  the 
same  The  strength  then  can  only  vary  according  to  the 
situation  of  the  centre  of  gravity  of  the  solid  part,  and  this 
again  will  depend  on  the  place  where  the  cylinder  must  first 
break,  or  on  the  manner  in  which  it  is  fixed. 

Now,  by  cor.  8 art.  251  Sta- 
tics, the  cylinder  is  strongest 
when  the  hollow,  or  inner  cir- 
cle, is  nearest  to  that  side 
where  the  fracture  is  to  end, 
that  is,  at  the  bottom  when  it 
breaks  first  at  the  upper  side, 
or  when  the  cylinder  is  fixed 
only  at  one  end  as  in  the  first 
figure.  But  the  reverse  will  be 
the  case  when  the  cylinder  is 
fixed  at  both  euds  : and  con- 
sequently when  it  opens  first  below,  or  ends  above,  as  in  the 
2d  figure  annexed. 

PROBLEM  33. 

To  determine  the  Dimensions  of  the  Strongest  Rectangular  Beam , 
that  can  be  cut  out  of  a Given  Cylinder. 

Let  ab,  the  breadth  of  the  required 
beam,  be  denoted  by  b,  ad  the  depth  by 
d,  and  the  diameter  ac  of  the  cylinder 
by  d.  Now  when  ab  is  horizontal,  the 
lateral  strength  is  denoted  by  bd 2 (by  art. 

248  Statics),  which  is  to  be  a maximum. 

But  ad2  = ac2 — AB2,ord2— d2 — b 2 ; 
theref.6d2=(D2  — b2)b=n2b  — 63isa  max- 
imum : in  fluxions  d2  b — 3 b2  b = 0 = d2  — 3 b2 , or  d2  ~ 5 b2  ; 
also  d2  —t>2  — b2  = 36  — b2  ~ 2 b2 . Conseq  b2  : d2  : d2  : : 
1:2:  3,  that  is,  the  squares  of  the  breadth,  and  of  the 
depth,  and  of  the  cylinder’s  diameter,  are  to  one  another 
respectively  as  the  three  numbers  1,  2,  3. 


Corel . 


516 


PROMISCUOUS  EXERCISES. 


Corol.  1.  Hence  results  this  easy  prac- 
tical construction  : divide  the  diameter  ac 
into  three  equal  parts,  at  the  points  e,  f ; 
erect  the  perpendiculars  eb.  fd  , and  join 
the  points  b,  d to  the  extremities  of  the 
diameter  : so  shall  abcd  be  the  rectangu- 
lar end  of  the  beam  as  required.  For, 
because  ae,  ab.  ac  are  in  continued  pro- 
portion (theor.  87  Geom.),  theref.  ae  : ac 
in  like  manner  af  : ac  : : ad2  : ac2 
ab  : ad2  : ac2  : : 1 : 2 : 3. 


hence  ae  : af  : ac 


Corol.  2.  The  ratios  of  the  three  b,  d,  d,  being  as  the 
three  1 , y/  2,  3,  or  as  1,  T414  1 -732,  are  nearly  as  the 

three  5,  7,  8 6,  or  more  nearly  as  12,  17,  20  8. 


Corol.  3.  A square  beam  cut  out  of  the  same  cylinder, 
would  have  its  side  = Pv/2“Idv/‘'  And  solidity  would 
be  to  that  of  the  strongest  beam,  as  4n2  to  |d2  y/  2,  or  as 
3 to  2^/2,  or  as  3 to  2 828  ; while  its  strength  would  be  to  that 
of  the  strongest  beam,  as  (D^/i)3  to  X f d2,  or  as  £ y/  2 
to  | v/3,  or  as  9^/2  to  8^3,  or  nearly  as  101  to  1 10. 

Corol.  4.  Either  of  these  beams  will  exert  the  greatest 
lateral  strength,  when  the  diagoual  of  its  end  is  placed  verti- 
cally by  art.  252  Statics. 

Corol.  5 The  strength  of  the  whole  cylinder  will  be  to 
that  of  the  square  beam,  when  placed  with  its  diagonal  ver- 
tically, as  the  area  of  the  circle  to  that  of  its  inscribed  square. 
For,  the  centre  of  the  circle  will  he  the  centre  of  gravity  of 
both  beams,  and  is  at  the  distance  of  the  radius  from  the 
lowest  point  in  each  of  them  ; conseq.  their  strengths  will 
be  as  their  areas,  by  art.  243  Statics. 


PROBLEM  34. 

To  determine  the  Difference  in  the  Strength  of  a Triangu- 
lar Beam , according  as  it  lies  ■with  the  Edge  or  with  the  Flat 
Side  Upwards. 

In  the  same  beam,  the  area  is  the  same,  and  therefore  the 
strength  can  only  vary  with  the  distance  of  the  centre  of 
gravity  from  the  highest  or  lowest  point  ; but  in  a triangle,  the 
distance  of  the  centre  of  gravity  from  an  angle;  is  double  of 
its  distance  from  the  opposite  side  : therefore  the  strength  of 
the  beam  will  be  as  2 to  1 with  the  different  sides  upwards, 
under  different  circumstances,  viz.  when  the  centre  of  gra- 
vity is  farthest  from  the  place  where  fracture  ends,  by  art  243 
Statics,  that  is,  with  the  angle  upwards  when  the  beam  is 

supported 


PROMISCUOUS  EXERCISES. 


517 


Supported  at  both  ends  ; but  with  the  side  upwards,  when  it 
is  ‘ ipported  only  at  one  end,  (art  252  Statics),  because  in  the 
former  case  the  beam  breaks  first  below,  but  the  reverse  in 
the  latter  case. 


PROBLEM  35. 

Given  the  Length  and  Weight  of  a Cylinder  or  Prism, 
placed  Horizontally  "with  one  end  firmly  fixed , and  will  just 
support  a given  weight  at  the  other  end  without  breaking  ; it 
is  required  to  find  the  Length  of  a Similar  Prism  or  Cylin- 
der whtch,  when  supported  in  like  manner  at  one  end  shall 
jus',  near  without  breaking  another  given  weight  at  the  unsupport- 
ed end. 

Let  l denote  the  length  of  the  given  cylinder  or  prism,  d 
the  diameter  or  depth  of  its  end,  w its  weight,  and  u the 
weight  hanging  at  the  unsupported  end  ; also  let  the  like 
capitals  l,  d,  w,  u denote  the  corresponding  particulars  of 
the  other  prism  or  cylinder.  Then,  the  weights  of  similar 
solids  of  the  same  matter  being  as  the  cubes  of  their  lengths, 

L 3 

as  l3  : l3  : : — w,  the  weight  of  the  prism  whose  length 

is  l.  Now  ±wl  will  be  the  stress  on  the  first  beam  by  its  own 
weight  w acting  at  its  centre  of  gravity,  or  at  half  its  length  ; 
and  lu  the  stress  of  the  added  weight  u at  its  extremity,  their 
sum  + u 'it  will  therefore  be  the  whole  stress  on  the  given 
beam  : in  like  manner  the  whole  stress  on  the  other  beam, 

whose  weight  is  w or  ~w,  will  be  (ivv+u)i.  or  + u)l. 

Rut  the  lateral  strength  of  the  first  beam  is  to  that  of  the 
second,  as  d3  to  n3  (art.  246  Statics),  or  as  ft  to  l3  ; and  the 
strengths  and  stresses  of  the  two  beams  must  be  in  the  same 
ratio,  to  answer  the  conditions  of  the  problem  ; therefore  as 

L 3 

(±w+u)l  : - - ^-(-u)l  ::  l3  : l3  ; this  analogy,  turned  into 


w -l~2u 


II2  ~l3  u — 0,  a cubic  equa- 


2l3 

an  equation,  gives  l3 

w w 

tion  from  which  the  numeral  value  of  l may  be  easily  deter- 
mined, when  those  of  the  other  letters  are  known. 

Corol.  1.  When  u vanishes,  the  equation  gives  l3  = 
w+Zu,  „ to  „ , 

L l2  , or  l = l whence  w : w-{-2 u : : l : l,  for  the 

W 'll) 

length  of  the  beam,  which  will  but  just  support  its  own 
weight. 

Corol.  2.  If  a beam  just  only  support  its  own  weight, 
when  fixed  at  one  end  ; then  a beam  of  double  its  length 
fixed  at  both  ends,  will  also  just  sustain  itself : or  it  the  one 
just  break,  the  other  will  do  the  same. 


PROBLEM 


518 


PROMISCUOUS  EXERCISES. 


PROBLEM  36. 

Given  the  Length  and  Weight  of  a Cylinder  or  Prism,  fixed 
Horizontally  as  in  the  foregoing  problem,  and  a weight  which, 
when  hung  at  a given  point.  Breaks  the  Prism  ; it  is  required 
to  determine  how  much  longer  the  Prism,  of  equal  Diameter  or 
of  equal  Breadth  and  Depth,  may  be  extended  before  it  Break, 
either  by  its  own  weight,  or  by  the  addition  of  any  other  adventi- 
tious weight. 

Let  l denote  the  length  of  the  given  prism,  w its  weight, 
and  w a weight  attached  to  it  at  the  distance  d from  the  fixed 
end  ; also  let  l denote  the  required  length  of  the  other  prism, 
and  u the  weight  attached  to  it  at  the  distance  d.  Now  the 
strain  occasioned  by  the  weight  of  the  first  beam  is  \wl,  and 
that  by  the  weight  u at  the  distance  d,  is  du,  their  sum  \wl 
-f  du  being  the  whole  strain.  In  like  manner  £wl  -f*  du  is 

the  strain  on  the  second  beam  ; butZ  : l : : w : ^ = w the 

££//2 

weight  of  this  beam,  theref.  + du  = its  strain  But  the 
strength  of  the  beam,  which  is  just  sufficient  to  resist  these 
strains,  is  the  same  in  both  cases  ; therefore  ~j-  + du  = 
hwl  -f-  du,  and  hence,  by  reduction,  the  required  length 
1=  ^ (l  X 

V)  1 

Carol.  1.  When  the  lengthened  beam  just  breaks  by  its 
own  weight,  then  u = 0 or  vanishes,  and  the  required  length 

, ,n  v tvl+Zdu. 

becomes  l — */(l  X ). 

v ' so 

Corol.  2.  Also  when  u vanishes,  if  d become  = l,  then 
l = l is  the  required  length. 

PROBLEM  37. 


Let  ab  be  a beam  moveable  about  the  end  a,  so  as  to 
■make  any  angle  bac  with  the  plane  of  the  horizon  ac  : it  is 
required  to  determine  the  position  of  a prop  or  supporter  de 
of  a given  length,  which  shall  sustain  it  with  the  greatest  ease 
in  any  given  position ; also  to  ascertain  the  angle  bac  when 
the  least  force  which  can  sustain  ab.  is  greater  than  the  least 
force  in  any  other  position. 


Let 


PROMISCUOUS  EXERCISES. 


519 


Let  g be  the  centre  of  gravity  of  the 
beam  ; and  draw  cm  perp.  to  ab,  g«  to 
ac,  nm  to  am,  and  afh  to  de.  Put 
t — AG,-  p — de,  w — the  weight  of  the 
beam  ab,  and  a n — x.  Then  by  the 
nature  of  the  parallelogram  of  forces, 
s n : got,  or  by  sim  triangles,  ag  = r : 

An  *=  x : : w •?— , the  force  which  acting 

at  g in  the  direction  me,  is  sufficient  to  sustain  the  beam  ; 


and  by  the  nature  of  the  lever,  ae  ? ag  = r 


the  re- 


quisite force  at  o : the  force  capable  of  supporting  it  at  e 


in  a direction  perp.  to  ab  or  parallel  to  me  ; and  again  as 
•WX  wx 
AE 


af  : ae  : : — : — , the  force  or  pressure  actually  sustained  by 

the  given  prop  de  in  a direction  perp.  to  af.  And  this  latter 
force  will  manifestly  be  the  least  possible  when  the  perp.  af 
upon  de  is  the  greatest  possible,  whatever  the  angle  bac  may 
be,  which  is  when  the  triangle  ade  is  isosceles,  or  has  the 
side  ad  = ae,  by  an  obvious  corol.  from  the  latter  part  of  prob. 
6,  Division  of  Surfaces,  vol.  1. 

Secondly,  for  a solution  to  the  latter  part  of  the  problem, 

we  have  to  find  when  — is  a maximum  ; the  angles  d and 
af 

e being  always  equal  to  each  other,  while  they  vary  in  mag- 
nitude by  the  change  in  the  position  of  ab  Let  af  produced 
meet  c»  in  h : then,  in  the  similar  triangles  adf,  ahti,  it 

will  be  af  : Are  = x : : df  = \p  : Hn,  hence  — = and 

^ af  ip 

conseq.  - X w = — X w.  But,  by  theor.  83  Geom.  and 
comp,  ag  + ah  = r + x : An  ~ x : : gn  = ✓ (r2-*2)  : 
h n = x / (r2  — x2)  — x r-~  : consequently  the  force 

^ X i»,  acting  on  the  prop,  is  also  truly  expressed  by 
2 P 

Then  the  'fluxion  of  this  made  to  vanish  gives 
==  the  cos.  angle  bac  = 51®  50',  the  inclination 

required. 


' PROBLEM 


520 


PROMISCUOUS  EXERCISES. 

PROBLEM  38. 


Suppose  the  Beam  ab,  instead  of  being  moveable  about  the 
centre,  a,  as  in  the  last  problem , to  be  supported  in  a given 
position  by  means  of  the  given  prop  de  : it  is  rec/uired  to  de- 
termine the  position  of  that  prop . so  that  the  prismatic  beum  ac 
on  which  it  stands,  may  be  the  least  liable  to  breaking,  this  latter 
beam  being  only  supported  at  its  two  ends  a and  c. 

Put  the  base  ac  = b , the  prop  de  = p, 
ag  — r,  the  weight  of  ab  = w,  s and  c the 
sine  and  cosine  of  x = sin  Z e, 

y — sin.  Z d,  and  ^ — ae.  Then,  by 

trigon.  z : y : : p : s,  or  — = 8 , and 

x p 

f)X 

ad  = — ; also  cw  = the  force  of  the  beam 

S 

at  g in  direction  am.  Let  f denote  the  force  sustaining  the 
beam  at  f.  in  the  direction  ed  : then,  because  action  and  re- 
action  are  equal  and  opposite,  the  same  force  will  be  exerted 
at  d in  the  direction  de  : therefore  ag  . cw  — fzx , and 

f = r-^.  Again,  the  vertical  stress  at  d,  will  be  as  f X sine 

D X AD  . DC  = Fy  . AD  . DC  = — ^ X (b  — — ) = (sub- 

zx  s v v 


stituting  - for  its  equal  -)  X — X — — — * = rcz»  X 

° P 7 z'  px  S S 

^ x ~ -C-W—  X (—  — x)  = a minimum  by  the  problem. 
Conseq.  — a:  is  a minimum,  or  xa  maximum,  that  is, 


x — 1,  and  the  angle  e is  a right  angle.  Hence  the  point  r 
is  easily  found  by  this  proportion,  sin.  a : cos.  a : : ed  : ea. 


PROBLEM  59. 


To  explain  the  Disposition  of  the  Parts  of  Machines. 

When  several  pieces  of  timber,  iron,  or  any  other  materials, 
are  employed  in  a machine  or  structure  of  any  kind,  all  the 
parts,  both  of  the  same  piece,  and  of  the  different  pieces  in 
the  fabric,  ought  to  be  so  adjusted  with  respect  to  magnitude, 
that  the  strengtli  in  every  part  may  be,  as  near  as  possible,  in 
a constant  proportion  to  the  stress  or  strain  to  which  they 
will  be  subjected  Thus,  in  the  construction  of  any  engine, 
the  weight  and  pressure  on  every  part  should  be  investigated 
and  the  strength  apportioned  accordingly.  All  levers,  for 
instance,  should  be  made  strongest  where  they  are  most 
strained  : viz.  levers  of  the  first  kind,  at  the  fulcrum  ; levers 


PROMISCUOUS  EXERCISES. 


521 


of  the  second  kind,  where  the  weight  acts  ; and  those  of  the 
third  kind,  where  the  power  is  applied.  The  axles  of  wheels 
and  pulleys,  the  teeth  of  wheels,  also  ropes,  &c.  must  be  made 
stronger  or  weaker,  as  they  are  to  be  more  of  less  acted  on. 
The  strength  allotted  should  be  more  than  fully  competent 
to  the  stress  to  which  the  parts  can  ever  be  liable  ; but  with- 
out allowing  the  surplus  to  be  extravagant  for  an  over  ex- 
cess of  strength  in  any  part,  instead  of  being  serviceable, 
would  be  very  injurious,  by  increasing  the  resistance  the  ma- 
chine has  to  overcome,  and  thus  encumbering,  impeding,  and 
even  preventing  the  requisite  motion  ; while,  on  the  other 
hand,  a defect  of  strength  in  any  part  will  cause  a failure 
there,  and  either  render  the  whole  useless,  or  demand  very 
frequent  repairs. 

PROBLEM  40. 

To  ascertain  the  Strength  of  V arious  Substances. 

The  proportions  that  we  have  given  on  the  strength  and 
stress  of  materials,  however  true,  according  to  the  principles 
assumed,  are  of  little  or  no  use  in  practice,  till  the  compara- 
tive strength  of  different  substances  is  ascertained  : and  even 
then  they  will  apply  more  or  less  accurately  to  different  sub- 
stances Hitherto  they  have  been  applied  almost  exclusively 
to  the  resisting  force  of  beams  of  timber  ; though  probably 
no  materials  whatever  accord  less  with  the  theory  than  timber 
of  all  kinds.  In  the  theory,  the  resisting  body  is  supposed 
to  be  perfectly  homogeneous,  or  composed  of  parallel  fibres, 
equally  distributed  round  an  axis,  and  presenting  uniform  re- 
sistance to  rupture.  Eut  this  is  not  the  case  in  a beam  of 
timber  : for,  by  tracing  the  process  of  vegetation,  it  is  readily 
seen  that  the  ligneous  coats  of  a tree,  formed  by  its  annual 
growth  are  almost  concentric  ; being  like  so  many  hollow 
cylinders  thrust  into  each  other,  and  united  by  a kind  of  me- 
dullary substance,  which  offers  but  little  resistance  : these 
hollow  cylinders  therefore  furnish  the  chief  strength  and 
resistance  to  the  force  which  tends  to  break  them. 

Now,  when  the  trunk  of  a tree  is  squared,  in  order  that  it 
may  be  converted  into  a beam,  it  is  plain  that  all  the  ligneous 
cylinders  greater  than  the  circle  inscribed  in  the  square  or 
rectangle,  which  is  the  transverse  section  of  the  beam,  are 
cut  off  at  the  sides  ; and  therefore  almost  the  whole  strength 
or  resistance  arises  from  the  cylindric  trunk  inscribed  in  the 
solid  part  of  the  beam  ; the  portions  of  the  cylindric  coats, 
situated  towards  the  angles,  adding  but  little  comparatively 
* to  the  strength  and  resistance  of  the  beam.  Hence  it  follows 
that  we  cannot,  by  legitimate  comparison,  accurately  deduce 

Vol,  II.  67  the 


522 


PROMISCUOUS  EXERCISES. 


the  strength  of  a joist,  cut  from  a small  tree,  by  experiments 
on  another  which  has  been  sawn  from  a much  larger  tree  or 
block  As  to  the  concentric  cylinders  above  mentioned,  they 
are  evidently  not  all  of  equal  strength  : those  nearest  the 
centre,  being  the  oldest,  are  also  the  hardest  and  strongest  ; 
which  again  is  contrary  to  the  theory,  in  which  they  are  sup- 
posed uniform 'throughout.  But  yet,  after  all  however,  it  is 
still  found  that,  in  some  of  the  mo-t  important  problems,  the 
results  of  the  theory  and  well-conducted  experiments  coin- 
cide, even  with  regard  to  timber  : thus,  for  example,  the  ex- 
periments on  rectangular  beams  afford  results  deviating  but 
in  a very  slight  degree  from  the  theorem,  that  the  strength 
is  proportional  to  the  product  of  the  breadth  and  the  square 
of  the  depth. 

Experiments  on  the  strength  of  different  kinds  of  wood, 
are  by  no  means  so  numerous  as  might  be  wished  : the  most 
useful  seem  to  be  those  made  by  Muschenbroek,  Buffon, 
Emerson,  Parent,  Banks,  and  Girard.  But  it  will  be  at  all 
times  highly  advantageous  to  make  new  experiments  on  the 
same  subject  ; a labour  especially  reserved  for  engineers  who 
possess  skill  and  zeal  for  the  advancement  of  their  profession. 
It  has  been  found  by  experiments,  that  the  same  kind  of 
wood,  and  of  the  same  shape  and  dimensions,  will  bear  or 
break  with  very  different  weights  : that  one  piece  is  much 
stronger  than  another,  not  only  cut  out  of  the  same  tree,  but 
out  of  the  same  rod  ; and  that  even,  if  a piece  of  any  length, 
planed  equally  thick  throughout,  be  separated  into  three  or 
four  pieces  of  an  equal  length,  it  will  often  be  found  that 
these  pieces  require  different  weights  to  break  them.  Emer- 
son observes  that  wood  from  the  boughs  and  branches  of  trees 
is  far  weaker  than  that  of  the  trunk  or  body  ; the  wood  of 
the  large  limbs  stronger  than  that  of  the  smaller  ones  ; and 
the  wood  in  the  heart  of  a sound  tree  strongest  of  all  ; though 
some  authors  differs  on  this  point.  It  is  also  observed  that  a 
piece  of  timber  which  has  borne  a great  weight  for  a short 
time,  has  broke  with  a far  less  weight,  when  left  upon  it  for 
a much,  longer  time  Wood  is  also  weaker  when  green,  and 
strongest  when  thoroughly  dried,  in  the  course  of  two  or 
three  years,  at  least.  Wood  is  often  very  much  weakened  by 
knots  in  it  ; also  when  cross-grained,  as  often  happens  in 
sawing,  it  will  be  weakened  in  a greater  or  less  degree,  ac- 
cording as  the  cut  runs  more  or  less  across  the  grain.  From 
all  which  it  follows,  that  a considerable  allowance  ought  to  be 
made  for  the  various  strength  of  wood,  w hen  applied  to  anyr 
use  where  strength  and  durability  are  required. 

Iron  is  much  more  uniform  in  its  strength  thaD  wood.  Yet 

experiment? 


PROMISCUOUS  EXERCISES. 


523 


r 

experiments  show  that  there  is  some  difference  arising  from 
different  kinds  of  ore  : a difference  is  also  found  not  only  in 
iron  from  different  furnaces,  but  from  the  same  furnace,  and 
even  from  the  same  melting  ; which  may  arise  in  a great  mea- 
sure from  the  different  degrees  of  heat  it  has  when  poured 
into  the  mould. 

Every  beam  or  bar,  whether  of  wood,  iron,  or  stone,  is 
more  eaily  broken  by  any  transverse  strain,  while  it  is  also 
suffering  any  very  great  compression  endways  ; so  much  so 
indeed  that  we  have  sometimes  seen  3,  rod,  or  a long  siender 
beam  when  used  as  a prop  or  shoar,  urged  home  to  sucli  a 
degree  that  it  has  burst  asunder  with  a violent  spring.  Se- 
veral experiments  have  been  made  on  this  kind  of  strain  : a 
piece  of  white  marble,  i of  an  inch  square,  and  3 inch.es  long, 
bore  381bs  ; but  when  compressed  endways  with  3001bs,  it 
broke  with  1 4^1bs.  The  effect  is  much  more  observable  in 
timber,  and  more  elastic  bodies  ; bat  is  considerable  in  all. 
This  is  a point  therefore  that  must  be  attended  to  in  all  ex- 
periments ; as  well  as  the  following,  viz  that  a beam  support- 
ed at  both  ends,  will  carry  almost  twice  as  much  when  the 
the  ends  beyond  the  props  are  kept  from  rising,  as  when  the 
beams  rest  loosely  on  the  props. 

The  following  list  of  the  absolute  strength  of  several  ma- 
terials, is  extracted  from  the  collection  made  by  professor 
Robison,  from  the  experiments  of  Muschenbroek  and  other 
experimentalists.  The  specimens  are  supposed  to  be  prisms 
or  cylinders  of  one  square  inch  transverse  area,  which  are 
stretched  or  drawn  lengthways  by  suspended  weights,  gradu- 
ally increased  till  the  bars  parted  or  were  torn  asunder  by  the 
number  of  avoirdupois  pounds,  on  a medium  of  many  trials, 
set  opposite  each  name. 

ij 

1st  Metals. 


lbs. 

lb. 

Gold,  cast  . . 

. 22,000 

Tin,  cast  .... 

5,000 

Silver,  cast  . . 

. 42,000 

Lead,  cast  .... 

860 

Copper,  cast  . . 

. 34,000 

Regulus  of  Antimony 

1,000 

Iron,  cast  . 

. 50,000 

Zinc 

2,600 

Iron,  bar  . 
Steel,  bar 

. 70,000 

. 135,000 

Bismuth  .... 

2,900 

. ■ . ' 

It  is  very  remarkable  that  almost  all  the  metallic  mixtures 
are  more  tenacious  than  the  metals  themselves.  The  change  of 
tenacity  depends  much  on  the  proportion  of  the  ingredients  ; 
and  yet  the  proportion  which  produces  the  most  tenacious 
mixture,  is  different  in  the  different  metals.  The  proportion 

of 


■ 524 


PROMISCUOUS  EXERCISES. 


of  ingredients  here  selected,  is  that  which  produces  the  great- 
est strength. 


2 parts  gold  with  1 

lbs. 

lbs. 

Brass,  of  copper  and  tin  51 ,000 

silver  .... 

28,000 

3 tin,  1 lead 

10,200 

5 pts  gold,  1 copper 

50,000 

8 tin,  1 zinc 

10,000 

5 silver,  1 copper  . 

48,5U0 

4 tin,  1 regul.  antim. 

12,000 

4 silver,  1 tin  . . 

41,000 

8 lead,  1 zinc  . 

4,500 

6 copper,  1 tin  . 

60,000 

4 tin,  1 lead,  1 zinc 

13,000 

These  numbers  are  of  considerable  use  in  the  arts.  The 
mixtures  of  copper  and  tin  are  particularly  interesting  in  the 
fabric  of  great  guns.  By  mixing  copper,  whose  greatest 
strength  does  not  exceed  37,000,  with  tin  which  does  not  ex- 
ceed 6000,  is  produced  a metal  whose  tenacity  is  almost  double, 
at  the  same  time  that  it  is  harder  and  more  easily  wrought : 
it  is  however  more  fusible.  We  see  also  that  a very  small 
addition  of  zinc  almost  doubles  the  tenacity  of  tin,  and  in- 
creases the  tenacity  of  lead  5 times  ; and  a small  addition  of 
lead  doubles  the  tenacity  of  tin.  These  are  economical  mix- 
tures ; and  afford  valuable  information  to  plumbers  for  aug- 
menting the  strength  of  water-pipes  Also,  by  having  re- 
course to  these  tables,  the  engineer  can  proportion  the  thick- 
ness of  his  pipes,  of  whatever  metal,  to  the  pressures  they 
are  to  suffer. 

2d.  Woods,  &c. 


lbs. 

lbs. 

Locust  tree  . 

. . 20,100 

Tamarind  . . . 

. 8,750 

Jujeb  . 

. . 18,500 

Fir  

. 8,330 

Beech, Oak 

. . 17,300 

Walnut  .... 

8,130 

Orange 

. . 15,500 

Pitch  pine  . . . 

. 7,650 

Alder  . 

. . 13,900 

Quince  .... 

6,750 

Elm 

. . 13,200 

Cypress  . . 

. 6,000 

Mulberry  . 

. . 12,500 

Poplar  .... 

. 5,500 

Willow  . . 

. . 12,500 

Cedar  .... 

. 4,880  ; 

Ash 

. . 12,000 

Ivory  .... 

. 16.270  | 

Plum  . . . 

. . 11,800 

Bone  .... 

..  5,250 

Elder  . . 

. . 10,000 

Horn  . . . 

. 8,750  | 

Pomegranate  . 

. . 9,750 

Whalebone. 

. 7,500  i 

Lemon . 

. . 9,250 

Tooth  of  sea-calf  . 

. 4,075  t 

It  is  to  be  observed  that  these  numbers  express  something  ; 
more  than  the  utmost  cohesion  ; the  weights  being  such  as 
will  very  soon  perhaps  in  a minute  or  two,  tear  the  rods  j 
asunder.  It  may  be  said  in  general,  that  | of  these  weights  I 
will  sensibly  impair  the  strength  after  acting  a considerable  : 
while,  and  that  one-half  is  the  utmost  that  can  remain  per- 
manently 1 


PROMISCUOUS  EXERCISES. 


525 


manently  suspended  at  the  rods  with  safety  ; and  it  is  this 
last  allotment  that  the  engineer  should  reckon  upon  in  his 
constructions.  There  is  however  considerable  difference  in 
this  respect  : woods  of  a very  straight  fibre,  such  as  fir,  will 
be  less  impaired  by  any  load  which  is  not  sufficient  to  break 
them  immediately.  According  to  Mr.  Emerson,  the  load 
which  may  be  safely  suspended  to  an  inch  square  of  various 
materials,  is,  as  follows  : 


lbs. 

lbs. 

Iron 

76,400 

Red  fir,  holly,  elder, 

Brass 

35,600 

plane  . 

. 

5,000 

Hempen  rope 

19,600 

Cherry,  hazle  . . 

4,760 

Ivory  

15,700 

Alder,  asp,  birch, 

Oak,  box,  yew,  plum 

7,850 

willow 

. 

4,290 

Elm,  ash,  beech  . . 

6,070 

Freestone 

, . . , 

914 

Walnut,  plumb  . . 

5,360 

Lead  . . 

430 

cwts. 

He  gives  also  the  practical  rule,  that 

Iron  . . . . 

135eZ2 

a cylinder  whose  diameter  is 

d inches, 

Good  rope 

22  cf2 

loaded  to  | of  its 

absolute 

strength, 

Oak  ...  . 

14d2 

will  carry  permanently  as 

here  an- 

Fir  . . . . 

9 d* 

nexed. 

Experiments  on  the  transverse  strength  of  bodies  are  easily- 
made,  and  accordingly  are  very  numerous,  especially  those 
made  on  timber,  being  the  most  common  and  the  most  in- 
teresting. The  completest  series  we  have  seen  is  that  given 
by  Belidor,  in  his  Science  des  Ingenieurs,  and  is  exhibited  in 
the  following  table.  The  first  column  simply  indicates  the 
number  of  the  experiments  ; the  column  b shows  the  breadth 
of  the  pieces,  in  inches  ; the  column  d contains  their  depths  ; 
the  column  l shows  the  lengths  ; and  column  lbs  shows  the 
weights  in  pounds  which  broke  them,  when  suspended  by 
their  middle  points,  being  the  medium  of  3 trials  of  each 
piece  ; the  accompanying  words,  fixed  and  loose  denoting 
.whether the  ends  were  firmly  fixed  down,  or  simply  lay  loose 
on  the  supports. 


No. 

6 

d 

l 

lbs. 

1 

1 

i 

18 

406 

loose. 

2 

1 

i 

18 

608 

fixed. 

3 

2 

i 

18 

805 

loose. 

4 

1 

2 

18 

1580 

loose. 

5 

1 

1 

36 

187 

loose. 

6 

1 

1 

36 

283 

fixed. 

7 

2 

2 

36 

1685 

loose. 

8 

12 

1 3 

2-L 

3 

36 

1660 

loose. 

By 


PROMISCUOUS  EXERCISES. 


•s  26 

By  comparing  experiments  1 and  3,  the  strength  appears 
proportional  to  the  breadth. 

Experiments  3 and  4 show  the  strength  to  be  as  the  breadth 
multiplied  by  the  square  of  the  depth. 

Experiments  1 and  5 show  the  strength  nearly  in  the  inverse 
ratio  of  the  lengths,  but  with  a sensible  deficiency  in  the  longer 
pieces. 

Experiments  5 and  7 show  the  strength  to  be  proportional  to 
the  breadth  and  the  square  of  the  depth. 

Experiments  1 and  7 show  the  same  thing,  compounded  with 
the  inverse  ratio  of  the  length  ; the  deficiency  of  which  is  net 
so  remarkable  here. 

Experiments  1 and  2,  and  experiments  5 and  6,  show  the  in- 
crease of  strength,  by  fastening  down  the  ends,  to  be  in  the  pro- 
portion of  2 to  3 ; which  the  theory  states  as  2 to  4,  the  dif- 
ference being  probably  owing  to  the  manner  of  fixing. 

Mr.  Buffon  made  numerous  experiments,  both  on  small 
bars,  and  on  large  ones,  which  are  the  best.  The  following 
is  a specimen  of  one  set,  made  on  bars  of  sound  oak,  clear  of 
knots. 


Length 

feet. 

Weight 

lbs. 

Broke 

with 

lbs. 

Bent 

inch. 

Time. 

min. 

7 

5350 

3 5 

29' 

j 56 

5275 

4-5 

22 

8 

< 68 

4600 

3 75 

15 

l 63 

4500 

4-7 

13 

9 

\ 77 

41t)0 

4-85 

14 

l 71 

3950 

5 5 

12 

10 

i 84 

3625 

5-83 

15 

} 82 

3600 

6 5 

15 

12 

\ 100 

3Q50 

7 

\ 

} 98 

2925 

8 

Column  1 shows  the  length  of  the  bar,  in  feet,  clear  be- 
tween the  supports. — Column  2 is  the  weight  of  the  bar  in 
lbs,  the  2d  day  after  it  was  felled. — Column  3 shows  the 
number  of  pounds  necessary  for  breaking  the  tree  in  a few 
minutes. — Col.  4 is  the  number  of  inches  it  bent  down  before 
breaking. — Col.  5 is  the  time  at  which  it  broke  — The  parts 
next  to  the  root  were  always  the  heaviest  and  strongest. 

The  following  experiments  on  othbr  sizes  were  made  in  the 
same  way  ; two  at  least  of  each  length  being  taken,  and  the 
table  contains  the  mean  results.  The  beams  were  all  squared, 
and  their  sides  in  inches  are  placed  at  the  top  of  the  columns, 

their 


PROMISCUOUS  EXERCISES. 


527 


their  lengths  in  feet  being  in  the  first  column.  The  numbers 
in  the  other  columns,  are  the  pounds  weight  which  broke  the 
pieces. 


4 

5 

6 

7 

8 

A 

7 

5312 

11525 

18950 

32200 

47649 

11525 

8 

4550 

9787 

15525 

26050 

39750 

10085 

9 

4025 

8308 

13150 

22350 

32800 

8964 

10 

3612 

7125 

11250 

19475 

27750 

8068 

12 

2987 

6075 

9100 

16175 

23450 

6723 

14 

5300 

7475 

13225 

19775 

5763 

16 

4350 

6362 

1 1000 

1H375 

5042 

18 

3700 

5562 

9245 

13200 

4482 

20 

3225 

4950 

8375 

11487 

4034 

22 

2975 

3667 

24 

2162 

3362 

28  j 

1775 

2881 

Mr.  Buffon  had  found,  by  many  trials,  that  oak  timber 
lost  much  of  its  strength  in  the  course  of  seasoning  or  drying  ; 
and  therefore,  to  secure  uniformity,  his  trees  were  all  felled 
in  the  same  season  of  the  year,  were  squared  the  day  after, 
and  the  experiments  tried  the  third  day.  Trying  them  in  this 
green  state  gave  him  an  opportunity  of  observing  a very"  cu- 
rious phenomenon.  When  the  weights  were  laid  quickly 
on,  nearly  sufficient  to  break  the  beam,  a very  sensible  smoke 
was  observed  to  issue  from  the  two  ends  with  a sharp  hissing 
sound  ; which  continued  all  the  time  the  tree  was  bending 
and  cracking  This  shows  the  great  effects  of  the  compres- 
sion, and  that  the  beam  is  strained  through  its  whole  length, 
which  is  shown  also  by  its  bending  through  the  whole 
length. 

Mr.  Buffon  considers  the  experiments  with  the  5-inch  bars 
as  the  standard  of  comparison,  having  both  extended  these  to 
greater  lengths,  and  also  tried  more  pieces  of  each  length. 
Now,  the  theory  determines  the  relative  strength  of  bars,  of 
the  same  section,  to  be  inversely  as  their  lengths  : but  most 
of  the  trials  show  a great  deviation  from  this  rule,  probably 
owing,  in  part  at  least,  to  the  weights  of  the  pieces  them- 
selves. Thus,  the  5-inch  bar  of  28  feet  long  should  have 
half  the  strength  of  that  of  14  feet  or  2650,  whereas  it  is 
only  1775  ; the  bar  of  14  feet  should  have  half  the  strength 
of  that  of  7 feet,  or  5762,  but  is  only  5300  ; and  so  of 
others.  The  column  a is  added,  to  show  the  strength  that 
each  of  the  5-inch  bars  ought  to  have  by  the  theorv. 

Mr. 


528 


PROMISCUOUS  EXERCISES. 


Mr.  Banks,  an  ingenious  lecturer  on  natural  philosophy 
has  made  many  experiments  on  the  strength  ot'  oak,  deal,  and 
iron.  He  found  that  the  worst  or  weakest  piece  of  dry  heart 
of  oak,  1 inch  square,  and  1 foot  long,  broke  with  602lbs,  and 
the  strongest  piece  with  974lbs  : the  worst  piece  of  deal  broke 
with  464lbs,  and  the  best  with  690lbs.  A like  bar  of  the 
worst  kind  of  cast  iron  2190lbs.  Bars  of  iron  set  up  in  posi- 
tions oblique  to  the  horizon,  showed  strengths  nearly  propor- 
tional to  the  sines  of  elevation  of  the  pieces.  Equal  bars 
placed  horizontally,  on  supports  3 feet  distant,  bore  cwt  ; 
the  same  at  2-i  feet  distance  broke  only  with  9 cwt. — An  arch- 
ed rib  ol  29±  feet  span,  and  11  inches  high  in  the  centre,  sup- 
ported 991  cwt  ; it  sunk  in  the  middle  3J  inches,  and  rose 
again  f on  removing  the  load.  The  same  rib  tried  without 
abutments,  broke  with  55  cwt. — Another  rib,  a segment  of  a 
circle,  291  feet  span,,  and  3 feet  high  in  the  middle  bore  1004 
cwt,  and  sunk  1T3¥  in  the  middle.  The  same  rib  without  abut- 
ments, broke  with  644  cwt. 

Mr  Banks  made  also  experiments  at  another  foundry,  on 
like  bars  of  1 inch  square,  each  yard  in  length  weighing  91bs, 


the  props  at  3 feet  asunder. 

The  1st  bar  broke  with 963  lbs. 

The  2d  ditto 958 

The  3d  ditto 994 

Bar  made  from  the  cupola,  broke  with  . . . 864 

Bar  equally  thick  in  the  middle,  but  the  ends 
shaped  into  a parabola,  and  weighed  6Ti  lbs, 
broke  with  874 


From  these,  and  many  other  experiments,  Mr.  Banks  con- 
cludes, that  cast  iron  is  from  31  to  4^  times  stronger  than 
oak  of  the  same  dimensions,  and  from  5 to  64  times  stronger 
than  deal. 

Some  Examples  for  Practice . 


The  theory,  as  has  been  before  mentioned,  is,  That  the 
strength  of  a bar.  or  the  weight  it  will  bear,  is  directly  as 
the  breadth  and  square  of  the  depth  divided  by  the  length. 
So  that,  if  b ^denote  the  breadth  of  a bar,  d the  depth,  l the 
length,  and  w the  weight  it  will  bear  ; and  the  capitals  b,  d, 
l,  w denote  the  like  quantities  in  another  bar  ; then,  by  the 

rule  : to  : : : w,  which  gives  this  general  equation 

bd2L\v  =bd2  Iw,  from  which  any  one  of  the  letters  is  easily 
found  when  the  rest  are  given. 

Now,  if  we  take,  for  a standard  of  comparison,  this  expe- 
riment of  Mr.  Banks,  that  a bar  of  oak  an  inch  square  and  a 

foot 


PROMISCUOUS  EXERCISES. 


52& 


Toot  in  length,  lying  on  a prop  at  each  end,  and  its  strengths 
or  the  utmost  weight  it  can  bear,  on  its  middle,  660lbs  : here 
b = lt  d — 1,1  — l,w—  660  ; these  substituted  in  the  above 
equation,  it  becomes  lw  = 660bd2,  from  which  any  one  of 
the  four  quantities  l,  w,  b,  d,  may  be  found  when  the  other 
three  are  given,  when  the  calculation  respects  oak  timber. 
But  for  fir  the  like  rule  will  be  lw  = 400bd2,  and  for  iron 
i,w  = 2640bd2. 

Exam,.  1.  Required  the  utmost  strength  of  an  oak  beam, 
of  6 inches  square  and  8 feet  long,  supported  at  each  end,  or 
the  weight  to  break  it  in  the  middle  ? 

Here  are  given  b = 6,  d = 6,  l = 8,  to  find  w = 660b° - 

— ^x6x:L6—  660  X 3 X 9 = 17820  lbs. 

8 

Exam.  2.  Required  the  depth  of  an  oak  beam,  of  the 
same  length  and  strength  as  above,  but  only  3 inches  breadth  ? 

Here,  as  3 : 6 ; : 36  : D2  = 72,  theref.  0 = ^72  = 8-485 
the  depth. 

This  last  beam,  though  as  strong  as  the  former,  is  but  little 
more  than  §■  of  its  size  or  quantity.  And  thus,  by  making 
joists  thinner,  a great  part  of  the  expence  is  saved,  as  in  the 
modern  style  of  flooring,  &c. 

Exam.  3.  To  determine  the  utmost  strength  of  a deal 
joist  of  2 inches  thick  and  8 inches  deep,  the  bearing  or 
breadth  of  the  room  being  12  feet  ? — Here  b = 2,  i>  = 8, 

l = 12  ; then  the  rule  lw  = 440bd2  gives  w = ~ 

440  X 2 X 6*  = 440X  32  = 4693  lbs. 

12  3 

Exam.  4.  Required  the  depth  of  a bar  of  iron  2 inchea- 
broad  and  8 feet  long,  to  sustain  a load  of  20.000lbs  ? — Here 
b = 2,  l = 8,  and  w = 20,000,  to  find  d from  the  equation 

lw  8x20000  1000 

lw  = 2640bd2  , viz.  d2  = — — = " = -4-—  30-3, 

’ 2640  b 2640X2  33  * 

and  d = ^/  30-3  = 51  inches  the  depth. 

Exam.  5.  To  find  the  length  of  a bar  of  oak,  an  inch 
square,  so  that  when  supported  at  both  ends  it  maj'  just  break 
by  its  own  weight  ? — Here  according  to  the  notation  and 
calculation  in  prob.  36,  l = 1,  w = § of  a lb,  the  weight  of 

1 foot  in  length,  and  u = 660lbs.  Then  l = l \/  ^ 2a  = 


.y/3301  = 57-45  feet,  nearly. 

Exam.  6.  To  find  the  length  of  an  iron  bar  an  inch  square, 
that  it  may  break  by  its  own  weight,  when  it  is  supported  at 
both  ends. — Here  as  before  l = 1,  w — 31bs  nearly  the 
Vbiv  II.  68  weight 


530 


PROMISCUOUS  EXERCISES. 


weight  of  1 foot  in  length,  also  u = 2640.  Therefore  l = 

l yw  -f  2u  _ ^ | £ee^  near]y. 

Note.  It  might  perhaps  have  been  supposed  that  this  last 
result  should  exceed  the  preceding  one  : but  it  must  be  con- 
sidered that  while  iron  is  only  about  4 times  stronger  than 
oak,  it  is  at  least  8 times  heavier. 

Exam.  7.  When  a weight  w is  suspended  from  e on  the 
arm  of  a crane  abcde,  it  is  required  to  find  the  pressure  at 
the  end  d of  the  spur,  and  that  at  b against  the  upright 
post  AC. 

C E 

Here,  by  the  nature  of  the  lever  --  w = 

the  pressure  at  d in  the  vertical  direction 
df  : but  this  pressure  in  df  is  to  that  in  db 

ce  ce  db 

as  df  to  db,  viz.  df  : db  : : w : w 

CD  df.cd 

the  pressure  in  db  ; and  again,  db  : fb  or  W 

CF..DB  CE  CE  . 

cd  : : w : — w = — w the  pressure 

Df.cd  df  BC 

against  b in  direction  fb. 

Thus,  for  example,  if  ce  = 16  feet,  bc  = 6,  cd  = 8, 

, ce  bd  1 6 • 10  „ 1(, 

bd  = 10,  and  w — 3 tons  ; then w g x,  o — ru 

’ BC  . CD  6.0 

C E 16 

tons  for  the  pressure  on  the  spur  db  Also  — w = -g-  x 
3=8  tons,  the  force  tending  to  break  the  bar  ac  at  b. 

PROBLEM  41. 


To  determine  the  circumstances  of  Space,  Penetration,  Velo- 
city, and  Time,  arising  from  a Ball  moving  with  a Given 
Velocity  and  Striking  a moveable  Block  of  Hood,  or  other  sub- 
stance. 


Let  the  ball  move  in  the  direction  ae  passing  through  the 
centre  of  gravity  of  the  block  b,  impinging  on  the  point  c : 
and  when  the  block  has  moved  through  the  space  cd  in 
consequence  of  the  blow,  let  the  ball  have  penetrated  to  the 

depth  de.  . , , . , 

Let  B = the  mass  or  matter  in  the  blocfc, 
b = the  same  in  the  ball, 
s = cd  the  space  moved  by  the  block. 


PROMISCUOUS  EXERCISES. 


531 


x — de  the  penetration  of  the  ball,  and  theref. 
s + x = ce  the  space  described  by  the  ball, 
a = the  first  velocity  of  the  ball, 
v = the  velocity  of  the  ball  at  e, 
u = velocity  of  the  block  at  the  same  instant, 
t = the  time  of  penetration,  or  of  the  motion, 
r = the  resisting  force  of  the  wood. 

Then  shall  - be  the  accelerating  force  of  the  block, 

and  y the  retarding  force  of  the  ball. 

0 

Now  because  the  momentum  Bu  , communicated  to  the 
block  in  the  time  },  is  that  which  is  lost  by  the  ball,  namely 
— bv  , therefore  b«  = — b'v  , and  b u — — bv.  But  when 
v — a,  u 0 ; therefore,  by  correcting,  b u =z  b(a  — v)  ; or 
the  momentum  of  the  block  is  every  where  equal  to  the  mo- 
mentum lost  by  the  ball  And  when  the  ball  has  penetrated 
to  the  utmost  depth,  or  when  u = v,  this  becomes  bw  — b 
( a — «,)  or  ab  — (b  + b)u ; that  is,  the  momentum  before 
the  stroke,  is  equal  to  the  momentum  after  it.  And  the  ve- 
locity communicated  will  be  the  same,  whatever  be  the  re- 
sisting force  of  the  block,  the  weight  being  the  same. 

Again,  (by  prob.  6,  Forces),  it  is  u2  — — rs~ , and  — 
v2  = 4-~  X (s  -f-  x),  or  rather,  by  correction,  a2  — v3  = 


4 Pfj* 

~-(s  -f-  x).  Hence  the  penetration  or  v 
b J 


>(■ 2?  — V2  ) — 4grs. 


b y r ~ r 4gr 

And  when  v = u,  by  substituting  u for  v,  and  b u2  for  4 grs, 
. , bu2—  (b+6)»(2  . . 

the  greatest  penetration  becomes — — - ; and  this  again 

by  writing  ab  for  its  value  (b  + b ) u,  gives  the  greatest  pene- 
tration x = y--— — -=^-X  (1  — ~r)‘  Which  is  barely 

equal  to  - — when  the  block  is  fixed,  or  infinitely  great  ; and 

is  always  very  nearly  equal  to  the  same-^—  when  b is  very 
great  in  respect  of  b. 


a-b : 


s2-f2p,b  a2  b 

A 


Hence  s -f-  x = - b = — b . . 

Atgr  4 gr  (B-tt?)2  4 gr 

And  theref.  b -f-  b : b + 2b  : : x : s -j-  x,  or  b + b : b : : x : $ 


, bx  nb2a2 

d s = = - — nr— 

B-f-6 

Exam.  When  the  ball  is  iron,  and  weighs  1 pound,  it 

penetrates 


53S 


PROMISCUOUS  EXERCISES. 


penetrates  elm  about  13  inches  when  it  moves  with  a velo- 
city of  1300  feet  per  second,  in  which  case, 

' a 2 15002  90002  i 

b 4 gr  4 X 16  tl  x 193X13  * 

When  b = 3001b,  and  6=1;  then  u — = 3 

b -j- b 501 

feet  nearly  per  second,  the  velocity  of  th€  block. 


Also  $ = 


B U2 


500  X 9 


„ , „ =— — r part  of  a foot,  or 

4gr  4 X 16  r^X  32284  461$  F 

of  an  inch,  which  is  the  space  moved  by  the  block  when  the 
ball  has  completed  its  penetration. 

And  t~~f~  iefe  = ek  part  of  a second>  or 
_2 ,26 

2s4-'2x  46!.}  1 12  6-f-13.231  1 , 

* = =-T?00-  = 6—  31  71500  = 692  part  °f  * ^ 

cond,  the  time  of  penetration 

PROBLEM  42. 

To  find  the  V elocity  and  Time  of  a Heavy  Body  descending 
down  the  Arc  of  a Circle , or  vibrating  in  the  Arc  by  a Line  fixed 
in  the  Centre. 

Let  d be  the  beginning  of  the  descent, 
c the  centre,  and  a the  lowest  point  of  the 
circle  ; draw  de  and  pq,  perpendicular  to 
ac.  Then  the  velocity  in  p being  the  same 
as  in  q.  by  falling  through  eq,  it  will  be 
•y=2v/(g  X eq)  = 8 y'  {a—x) , when  o=ae, 
cc  = AQ. 


But  the  flux,  of  the  time  i is  = 


-,  and  ap 


v'  (2 rx  — x2) 

— x 


where  r = the  radius  ac.  Theref.  I = - x r, 

8 </(2rX-x*X  y{a—x) 


_ ZTJi 

16  ^/(ax-ar2)Xv'(‘/-'r) 


where  d = 2r  the  diameter. 


_ - -yd 
16 


v'(ax-iJ)Xv'(l-'^) 


rt  16  X ^/{ax-xs')^  ^~2d~r2 


OJ"2  , 1 

~4d2*~  2 


.5X3 

;3&c-)> 


. 4.  6u3 


by  developing  ( 1 — -)  in  a series. 


But  the  fluent  of- 


, is  ~ X arc  to  radius  4a  and 

V(ax-I2)  a 

vers,  x,  or  it  is  the  arc  whose  rad.  is  1 arft  vers.  : which 

call  a And  let  the  fluents  of  the  succeeding  terms,  without 
the  coefficients,  be,  b,  c,  d,  e,  &c.  Then  will  the  fluxion  of  any 

one 


PROMISCUOUS  EXERCISES. 


5,33 


one,  as  q_,  at  n distance  from  a,  be  4.  = xn  a = xp,  which 
suppose  also  = the  flux,  of  bp  — dxn~l  (ax—x2)  = b p — 

d(n-l)ixn~2y  (ax-x2)-dxxn~2  X = b*  ~ 

dx  X — — !!£L  = 6p  — d(n— ^)ai>-f-  dnjp. 

Hence,  by  equating  the  coefhcients  of  the  like  terms. 


1 ; & 


271  — 1 


277 


a ; and  u 


(2ti  — i)ap  — 2xn~  i — x2) 

272 


Which  being  substituted,  the  fluential  terms  become  ~ X 

1 6 

, 1 aA  — 2^/(.tx— x2)  1.3  3aB  — 2ary  (“-  Jr2 ) 

a — — • ^ 

1.3.5  5ac  — 2ar2v'(«A7 — x2) 


2 .42/2 


2 . 4 • 6rf3  6 

will  be  found  by  art.  80  Fluxions. 


&c).  Or  the  same  fluents 


But  when,  x — a,  those  terms  become  barely  x 


16 


&c)  ; which  being 


12a  12.32a2  12. 32.52a2 

^ — ~ ~d  ~ 22 . 4 2 2 22.42.62^3 

subtracted,  and  x taken  = 0,  there  arises  for  the  whole 
time  of  descending  down  da,  or  the  corrected  value  of  t ~ 
3*141 6^/d  v „ ,t  i l2a  , 12.32  a2 


x (i  + 


4-  , . 

16  v ! 2 2d  22.42^2  1 22.42-62(/3 

When  the  arc  is  small,  as  in  the  vibration  of  the  pendu- 
lum of  a clock,  all  the  terms  of  the  series  may  be  omitted 
after  the  second,  and  then  the  time  of  a semi-vibration  t is 


12  .32^3 

A2,/o  ‘ J 


. 1.5708  r . a , 

nearly  = — — ~ x (1  + — ). 


And  theref  the  times  of 


vibration  of  a pendulum,  in  different  arcs,  are  as  8r  + «,  or  8 
times  the  radius  added  to  the  versed  sine  of  the  arc. 

If  d be  the  degrees  of  the  pendulum’s  vibration,  on  each 
side  of  the  lowest  point  of  the  small  arc,  the  radius  being  r, 
the  diameter  d,  and  3-1416  =p  : then  is  the  length  of  that 

But  the  versed  sine  in  terms  of  the 

A« 


arc  a 


pm  pdr> 

: 180  ~ 360' 


arc  is  a ■. 


— — + &c.  = — 

2r  24r2  d od3 


4-  &c. 


Therefore 
P2  d2 


a A2  /)2D2  p4D4 

— h &c.  — — -4-  &c.  or  onlv 

d d2  3di  ^ 3602  3‘3604  y 3602 

the  first  term,  by  rejecting  all  the  rest  of  the  terms  on  ac- 

npanv  ^ 

27- 


count  of  their  smallness,  or  - = nearly 

d 9>-  J 


This 


value  then  being  substituted  for  - or  — 

° d 2r 


in 


value  of  the  time,  it  becomes  t = 


1-5708 

4 


13131 
the  last  near 

D2 


^ s.x  + 

nearly. 


3 


PROMISCUOUS  EXERCISES. 


nearly.  And  therefore  the  times  of  vibration  in  different 
small  arcs,  are  as  52524  + d2  , or  as  52524  added  to  the  square 
of  the  number  of  degrees  in  the  arc. 

Hence  it  follows  that  the  time  lost  in  each  second,  by  vi- 
brating in  a circle,  instead  of  the  cycloid,  is  - ; and  con- 
sequently the  time  lost  in  a whole  day  of  24  hours,  or  24  X 
60  X 60  seconds,  is  f d2  nearly.  In  like  manner,  the  seconds 
lost  per  day  by  vibrating  in  the  arc  of  A degrees,  is  ^ a2. 
Therefore,  if  the  pendulum  keep  true  time  in  one  of  these 
arcs,  the  seconds  lost  or  gained  per  day,  by  vibrating  in  the 
other,  will  be  f (d2  — A2).  So,  for  example,  if  a pendulum 
measure  true  time  in  an  arc  of  3 degrees,  it  will  lose  1 is.  se- 
conds a day  by  vibrating  4 degrees  ; and  26|  seconds  a day  by 
vibrating  4 degrees  ; and  so  on. 

And  in  like  manner,  we  might  proceed  for  any  other  curve, 
as  the  ellipse,  hyperbola,  parabola.  &c 

Scholium.  By  comparing  this  with  the  results  of  the  pro- 
blems 13  and  14,  Prac.  Ex  on  Forces,  it  will  appear  that  the 
times  in  the  cycloid,  and  in  the  arc  of  a circle,  and  in  any  chord 
of  the  circle,  are  respectively  as  the  three  quantities. 


1,  1 + — &c  and — — 
8r  7 854 


or  nearly  as  the  three  quantities  1,1+  1-27324;  the  first 


and  last  being  constant,  but  the  middle  one,  or  the  time  in  the 
circle,  varying  with  the  extent  of  the  arc  of  vibration.  Also 
the  time  in  the  cycloid  is  the  least,  but  in  the  chord  the 
greatest  ; for  the  greatest  value  of  the  series,  in  this  prob. 
when  a = r,  on  the  arc  ad  is  a quadrant,  is  1 18014  ; and  in 
that  case  the  proportion  of  the  three  times  is  as  the  numbers 
1,  1-18014,  1-27324.  Moreover  the  time  in  the  circle  ap- 
proaches to  that  in  the  cycloid,  as  the  arc  decreases,  and  they 
are  very  nearly  equal  when  that  arc  is  very  small. 


PROBLEM  43. 

To  find  the  time  and  Velocity  of  a Chain,  consisting  of  very 
small  links,  descending  from  a smooth  horizontal  plane  ; the 
Chain  being  100  inches  long,  and  one  inch  of  it  hanging  off  the 
Plane  at  the  commencement  of  Motion. 

Put  a = l inch,  the  length  at  the  beginning  ; 
l — 100  the  whole  length  of  the  chain  ; 
x = any  variable  length  of  the  plane. 

Then  x is  the  motive  force  to  move  the  body, 

and  % — f the  accelerative  force. 


Hence 


PROMISCUOUS  EXERCISES. 


535 


Hence  vv  = %/s  = % X ~Xx  = 

• 2^*3?  ^ 

The  fluents  give  v 2 = — - — . But  v = 0,  when  x = a, 

theref  by  correction  v2  — % X -■  , and  i’=A/(%  X * ~a  ) 

the  velocity  for  any  length  x.  And  when  the  chain  just 
quits  the  plane,  x = 1,  and  then  the  greatest  velocity  is 

y/  (%X^L)  = v/  (2X193  X = 

196-45902  inches,  or  16-371585  feet,  per  second. 

* l 'x 

Again  \ or  i = <V  — X — — -- — — ; the  correct  fluent  of 

v 2§-  v/  x2“"u  ) 

Z /fx2—a,2) 

which  is  i = — X log.  — — , the  time  for  any 

103 

length  x • And  when  x = Z = 100,  it  is  t = ^ X log. 

100  + s/9999  _ 2-09676  seconds,  the  time  when  the  last  of  the 
chain  just  quits  the  plane. 

PROBLEM  44. 


To  find  the  Time  and  Velocity  of  a Chain , of  very  small 
Links , quitting  a Pulley,  by  passing  freely  over  it : the  whole 
Length  being  200  Inches . and  the  one  End  hanging  2 Inches  be- 
low the  other  at  the  Beginning. 

Put  a = 2,1  = 200,  and  x = bd  any  variable 
difference  of  the  two  parts  ab,  ac.  Then 


f = />  an<3  vv  or  2gfs  = 2g  . ^ i 


A 

G 


,X2. 


and 


Hence  the  correct  fluent  is  v2  = gX- 

-s  = s/  % X*^ — y— — ),the  general  expression  for  the 
veloc.  And  when  x ==  or  when  c arrives  at  a,  it 
is-  = v/feX^)  = s/(193X^3!)  = 

V (3B6  X1^)=y3i^L9  = 196-45902 

inches,  or  16-371585  feet  for  the  greatest  velocity 
when  the  chain  just  quits  the  pulley. 


D 


E 


A 

0 


C 


D 


Again,  t ot-  = ^-— y/~ X — . 

v 2v  v 4 g — a 2 ) 

J . . J?+v/(X2— <j3) 


rect  fluent  is  t 
expression  for  the  time 


v/  - X log. 


And  the  cor- 

the  general 
And  when  x = /,  it  become  < = 

y/ 


536 


PROMISCUOUS  EXERCISES. 


✓£x  log.  v = x i^gs+^w^-g, 

v'  X log.  — = 2 69676  seconds,  the  whole 

JOO  1 

time  when  the  chain  just  quits  {he  pulley. 

So  that  the  velocity  and  time  at  quitting  the  pulley  in  this 
prob.  and  the  plane  in  the  last.prob  are  the  same  ; the  dis- 
tance descended  99  being  the  same  in  both  For  though 
the  weight  l moved  in  this  latter  case,  be  double  of  what  it 
was  in  the  former,  the  moving  force  x is  also  double,  because 
here  the  one  end  of  the  chain  shortens  as  much  as  the  other 
end  lengthens,  so  that  the  space  descended  ix  is  doubled, 

and  becomes  x ; and  hence  the  accellerative  force  - of  f is 

the  same  in  both  ; and  of  course  the  velocity  and  time  the 
same  for  the  same  distance  descended. 


PROBLEM  45. 


A 


To  find  the  Number  of  Vibrations  made  by  two  Weights, 
connected  by  a very  fine  Thread , passing  freely  over  a Tack  or 
a Pulley,  while  the  less  Weight  is  drawn  up  to  it  by  the  Descent 
of  the  heavier  Weight  at  the  other  End. 

Suppose  the  motion  to  commence  at  equal  dis- 
tances below  the  pulley  at  b ; and  that  the  weights 
are  1 and  2 pounds 

Put  a = ab,  half  the  length  of  the  thread  ; 

b = 39|  inc.  or  3ff  feet,  the  second’s  pend, 
x = bw  = bw,  any  space  passed  over  ; 
z = the  number  of  vibrations. 

Then  '—T’  ~ f — i js  the  accelerating  forc-e. 
w-J-w  J 

And  hence  sor^/  4 gfs  — v / 4 gfx,  and  t or  — = - 


ICO 


33  i 


V^sJx 

But  by  the  nature  of  pendulums,  (a±x)  : ^/b  : : 1 vibr.  ; 

the  vibrations  per  second  made  by  either  weight, 

namely,  the  longer  or  shorter,  according  as  the  upper  or 
under  sigrris  used,  if  the  threads  were  to  continue  of  that 
length  for  1 second.  Hence,  then,  as 

i':i  -.i  i = i 

v a±;x  v azLT 

the  fluxion  of  the  number  of  vibrations. 

Now  when  tiie  upper  sign  + takes  place,  the  fluent  is 

_ „ , t>  (a+*)_  / 6 s/ 1 01-2x4.2 </(ax+x2) 

v 4 gf  v 4 gf  a 

and 


4 gf  v'Cax  ± i2,) 


PROMISCUOUS  EXERCISES. 


537 


And  when  x — a,  the  same  then  becomes  z = 1/  — - X W, 

Sf  S 

1 + a/  2 = v/^r  Xlog.  1 + v/2=a/7^ xl°g-  1 + a/2= 
"688511,  the  whole  number  of  vibrations  made  by  the  descend- 
ing weight. 

But  when  the  lower  sign,  or  — , takes  place,  the  fluent  is 
b 2 jo 

a/  — % X arc  to  rad.  1 and  vers.  — . Which,  when  x — a, 
v 4 gf  , a • 

. y 6 q ^ , n ,3x  :!9i  3-1416  v 117f 

s,res ^ jr  1416 = — rd  = 

1-227091,  the  whole  number  of  vibrations  made  by  the  lesser 
or  ascending  weight. 

Schol.  It  is  evident  that  the  whole  number  of  vibrations, 
in  each  case,  is  the  same,  whatever  the  length  of  the  thread 
is.  And  that  the  greater  number  is  to  the  less,  as  1-5708  to 
the  hyp.  log.  of  1 + ^/2. 

Farther,  the  number  of  vibrations  performed  in  the  same 
time  t,  by  an  invariable  pendulum,  constantly  of  the  same 

length  a,  is  ~ = -781190.  For,  the  time  of  descending 


Sf 

the  space  a,  or  the  fluent  of  t ~ 


■S^gfx' 


when  x = a,  is  t ==■ 


— . And,  by  the  nature  of  pendulums,  a : b : : 

Sf 

1 vibr.  : the  number  of  vibrations  performed  in  1 se- 

b b b 

cond  ; hence  1"  : t : : */  - : t */-  = ./  — . the  constant  num- 
v a v a sf 

her  of  vibrations. 

So  that  the  three  numbers  of  vibrations,  namely,  of  the 
ascending,  constant,  and  descending  pendulums,  are  propor- 
tional to  the  numbers  1-5708,  1,  and  hyp.  log.  1 -f-  2,  or  as 

1-5708,  1,  and  -88137  ; whatever  be  the  length  of  the  thread. 


REMARK, 


The  solution  here  given  by  Dr.  Hutton  to  this  45th  pro- 
blem, is  erroneous  ; one  of  his  errors  in  the  solution  consists 
in  his  not  attending  to  the  difference  of  tension  in  the  pendu- 
lum as  it  ascends,  descends,  or  eoptinues  of  an  invariable 
length  ; his  method  will  give  vibrations  to  the  descending 
pendulum,  even  when  the  tension  is  infinitely  small  or  noth- 
ing. A true  investigation  of  the  problem  affords  several  cu- 
rious results  ; but  in  some  cases  we  are  led  to  very  tedious 
computations. 

Vox,.  IF.  69 


PROBLEM 


338 


PROMISCUOUS  EXERCISES. 

PROBLEM  46. 


To  determine  the  Circumstances  of  the  Ascent  and  Descent 
of  two  unequal  Weights , suspended  at  the  two  Ends  of  a 
Thread  passing  over  a Pulley  : the  Weight  of  the  Thread 
and  of  the  Pulley  being  considered  in  the  Solution. 


W 


A 


ez* 


Let  l = the  whole  length  of  the  thread; 
a — the  weight  of  the  same  ; 
b = aw  the  dif  of  lengths  at  first  ; 
d — w — w the  dif  of  the  two  weights  ; 
e = a weight  applied  to  the  circumference, 
such  as  to  be  equal  to  its  whole  wt.  and 
friction  reduced  to  the  circumference  ; 
s = w+®+a+c  the  sum  of  the  weights  moved. 

Then  the  weight  of  b is  e~,  and  d . is  the  moving  force 

at  first.  But  if  x denote  any  variable  space  descended  by  w, 
or  ascended  by  w,  the  difference  of  the  lengths  of  the  thread 
will  be  altered  2x  ; so  that  the  difference  will  then  be  b—2x, 

and  its  weight  - a ; conseq.  the  motive  force  there  will  be 


d — 


b—2x  dl — nb+2ax 


and  theref. 


dl — ab+2ar 


= f the  ac- 


l l tl 

tolerating  force  there.  Hence  then  v'v  = %gfx  = 2gx  X 
dl — ab  4-2&X  , _ r , . . dl—cb~puB 

; the  fluents  of  which  give  o3  = 4 gx  X- 


a 1 — ® *1 

Or  v *=  2 y/  ^ X 4^/  (ex+x2)  the  general  expression  for  the 


velocity,  putting  e = 


dl-ab 


And  when  x = b,  or  w becomei 


as  far  below  w as  it  was  above  it  at  the  beginning,  it  is  barely 


v = 2 y/  — for  the  velocity  at  that  time.  Also,  when  a. 


the  weight  of  the  thread,  is  nothing,  the  velocity  is  only 


2 y/  l—,  as  it  ought. 


Again,  for  the  time  / or  = l «/  — X — — ; the 

fluents  of  which  give  t = */  — X log.  ■V'J— ^ the  ge- 

US  ^ e 

neral  expression  for  the  time,  of  descending  any  space  x. 

And  if  the  radicals  be  expanded  in  a series,  and  the  log. 
of  it  be  taken,  the  same  time  will  become 

sx  dl  r x*  i "3x's  . . 


dg  " v dl — ab  K * 6e 

Which  therefore  becomes  barely  y/ ^ when  a,  the  weight 
of  the  thread,  is  nothing  ; as  it  ought.  PROBLEM 


PROMISCUOUS  EXERCISES. 


539 


PROBLEM  47. 

To  find  the  Velocity  and  Time  of  Vibration  of  a small  Weight, 
fixed  to  the  ^middle  of  a Line,  or  fine  Thread  void  of  Gravity, 
v and  stretched  by  a given  Tension ; the  extent  of  the  Vibration 
' being  very  small. 


d 


D 


Let  l — ac  half  the  length  of  the  thread  ; 
a = cd  the  extent  of  the  vibration  ; 
x — ce  any  variable  distance  from  c ; 

•w  = wt  of  the  small  body  fixed  to  the  middle  ; 
w = a wt.  which  hang  at  each  end  of  the  thread,  will 
be  equal  to  the  constant  tensioaat  each  end,  acting 
in  the  direction  of  the  thread. 

Now,  by  the  nature  of  forces,  ae  : ce  : : w the  force  in 
direction  ea  : the  force  in  direction  ec.  Or,  because  ac  it 
nearly  = ae,  the  vibration  being  very  small,  taking  ac  in- 

stead  of  ae,  it  is  ac  : ce  : : w : — J-  the  force  in  ec  arising 

from  the  tension  in  ea.  Which  will  be  also  the  same  for 

that  in  eb.  Therefore  the  sum  is  — the  whole  motive 

force  in  ec  arising  from  the  tensions  on  both  sides.  Conse- 

2 ’vse 

quently  - — = f the  accelerative  force  there.  Hence  the 

equation  of  the  fluxions  v-i  or  2 gfs  = — ; and  the  flus, 

v2  = — — . But  when  x = a,  this  is  — 1^— — and  should 

Iso  lw 

Q 2 — 

be  = 0 ; theref.  the  correct  fluents  are  v2  — 4 gw  X , — , 

° lw 

and  hence  v = ^ (4gw  X ° ) the  velocity  of  the  little 

ff<iy 

body  w at  any  point  e.  And  when  x — o,  it  is  v = 2a  y/— 


for  the  greatest  velocity  at  the  point  c. 

Now  if  we  suppose  zv  = 1 grain,  w = 5lb  troy,  or  28800 
grains,  and  21  = ab  = 3 feet  ; the  velocity  at  c becomes. 


a ^ 


8xl6T2-X'2SSo.; 

3 


llllfa. 


So  that 


if  a = tl  inc.  the  greatest  veloc.  is.  9T5y  ft.  per  sec. 

if  a =?  1 inc.  the  greatest  veloc.  is  92 ft.  per  sec. 

if  a = 6 inc.  the  greatest  vel«c,  is  555TVJt.  per  sep.  To 


540 


PROMISCUOUS  EXERCISES. 


lw 


To  find  the  time  t,  it  is  t or  — = \ J — X , , a 

v v vrg  y/  (a*  - Xi) 

Hence  the  correct  fluent  is  t = 4 y/  — X arc  to  cosine  — and 

0 

radius  1,  for  the  time  in  de.  And  when  x =i=  0,  the  whole 

ivl 

time  in  dc,  or  of  half  a vibration,  is  *7854  y/  — ; and  conseq. 

ajjl 

the  time  of  a whole  vibration  through  d d is  1-5708  y/  — . 

Using  the  foregoing  numbers,  namely  w = 1,  w = 28800. 
and  2 1 — 3 feet  ; this  expression  for  the  time  gives 


_ 


3-1416 

353f , the  number  of  vibrations  per  second.  But  if  w — 2. 
there  would  be  250  vibrations  per  second  ; and  if  w = 100, 
there  would  be  35|f  vibrations  per  second. 

PROBLEM  48. 


To  determine  the  same  as  in  the  last  Problem,  when  the  Dis- 
tance cd  bears  some  sensible  Proportion  to  the  Length  ab  ; the 
Tension  of  the  Thread  however  being  still  supposed  a Constant 
Quantity. 


d 


D 


Using  here  the  same  notation  as  in  the  last  problem,  and 
taking  the  true  variable  length  ae  for  ac,  it  is  ae  or  eb  : ce  : : 

2w  : — — 77— the  whole  motive  force  from  the  two 

ae  vU2  + jt2) 

. 2w  x 

eaual  tensions  w in  ae  and  eb  ; and  theref.  — X f 

4 TO  v/(!2  + x2)  J 

is  the  accelerative  force  at  e.  Theref.  the  fluxional  equation 

is  v'v  or  2 gfl  = ^X—^ 5 and  the  fluents  tt2=— ^X 
J rv  to 

— yf  [l-  x2).  But  when  x — a,  these  are  0 = X — 

(l2  + a2)  ; therefore  the  correct  fluents  are  v2  = ^-^X 

r v ' TO 

W (J2  + ft=)  ~V  V2  + x2)]  = 8~  X (ad  — ae).  And 

hence  v = y/  [— - X (ad  — ae)]  the  general  expression  for 

the  velocity  at  e.  And  when  e arrives  at  c,  it  gives  the 

greatest 


* 


PROMISCUOUS  EXERCISES. 


541 


greatest  velocity  there  = x (ad  — ac)].  Which 

when  w = 28800,  w — 1,  21  ~ 3 feet,  and  cd  = 6 inches 


or  \ a foot,  is  (8X28800  X 16TVX 


x/10-3  _ 


54Si  feet 


per  second.  Which  came  out  555r\  in  the  last  problem,  by 
using  always  ac  for  ae  in  the  value  of  f.  But  when  the  ex- 
tent of  the  vibrations  is  very  small,  as  7l  of  an  inch,  as  it 
commonly  is,  this  greatest  velocity  here  will  be  8X28800 
X 16-J^  X 43J00  — 9 1 nearly,  which  in  the  last  problem  was 
9tSo  nearly. 

To  find  the  time,  it  is  } or — — — ~ X — ? — — , 

v v S\\ff  V’[c—  v'(/i+x2)}’ 

making  c = ad  = ^/(I2  + ft2)  To  find  the  fluent  the  easier, 
multiply  the  numer.  and  denom.  both  by  •v/[c-rv/G2  +'rZ);] 

so  shall  i = a/  —g  X x2j  X y/  [c  + ,/( J 2 + **)]• 

Expand  now  the  quantity  [c  -f-  (Z2  + x2)]  in  a series, 

and  put  d ==  c -j-  l,  so  shall;  = X , — -(I+77,— 

r v 8w g </(a2—x2y  4 at 


2d  + l„A  -f  2dl+ l2  40d3+M2l+mdP+5l3 


2048 


zs&c)  Now 


is  = the  arc  to  sine  — 


Z2d2~l3X  * 128^3/5 

the  fluent  of  the  first  term  m Jl 

VCa2  — x2)  a 

and  radius  1,  which  arc  call  a ; and  let  p,  q be  the  fluents  of 
any  other  two  successive  terms,  without  the  coefficients,  the 
distance  of  0.  from  the  first  term  a being  n ; then  it  is  evi- 
dent that  q_=  x2  r = x2nA,  and  p ==  x2”-2  a Assume  theref. 
q •&=  bp — ex2n~ 1 v^(a2  — -x2)  ; then  is  q^ot  x 2 p —b  p — (2 n — 1) 
. ex2nl,  ("2 n — IV.^- 

ex'2”-2  x \/(a2  x2~)  "4- — — — — b p— . 

(2  n — 


- + 


exin'x 


*/(, a2  ~x2)  ' ^/{u2 —x2) 

- «.)  • ^-WrJp-(2il“1>a2  r+(2rc-l)e*2  p + 

ex2P  = fci.  — (2n—  1)  ea2p  -f-  2?iex2i>.  Then,  comparing  the 
coefficients  of  the  like  terms,  we  find  1 = 2 en,  and  b — 

from  which  are  obtained  e = — ,andf>  — — — - a2. 

2 n 2« 


(2n—  l)ea 
Consequently  q 


(2 n ~ 1 )a2  p — ar2«— i (a2  — x2) 


2n 


,the  general 


equation  between  any  two  successive  terms,  and  by  means  of 
which  the  series  may  be  continued  as  far  as  we  please.  And 
hence,  neglecting  the  coefficients,  putting  a = the  first  term, 

namely  the  arc  whose  sine  is  -y,  andB,  c,d,  &c  the  follow- 
ing terms,  the  series  is  as  follows,  a -f*  ‘i-Jl—iX'— — _dL_'-|_ 

3u2b 


542 


PROMISCUOUS  EXERCISES. 


3a2  B —a:3  v/(a: 


2 _a;2 


5j2  c — x5  v/(a2  — ar2  ) 


&c.  Now  when  x=B, 

«JJ  O 

this  series  = 0 ; and  when  x = a,  the  series  becomes  \p  + 
&c  where  p = 3-1416,  or  the  series  is 


L . J a-  B - J 

+ — r +- 


1.3.5 


ae  &c  ) 


^(1  +ia2  + 2~4  a ■ 2.4.6' 

So  that,  by  taking  in  the  coefficients,  the  general  time  of  pass- 
ing over  any  distance  de  will  be 


^rl±JlX  ip  X (1  + - 
V tiwg  ' ' Adi 


2d  + l 


„ , - „ — o4&c.-arc  sin. 

32^2/s  2 . 4 

q2  a — — a;2 ) 2 d -f-  1 ofl2B-i3^(u; 


&C. 


1 

a Adi  ' 2 ‘ 32;.2/3  4 

And  hence,  taking  x = 0,  and  doubling,  the  time  of  a 
•whole  vibration,  or  double  the  time  of  passing  over  cd  will 


foe  equal  to  X (1  + ^ 

4J2  4 2e//4-  /z  1 . 3 


S' 


, 2d  + i 1 • 3 . , 

a2 . a4  + 

32d3  (32.4 


l'2Sd*V  *2.4.6 


2wg  v ' 4d/ 

.5  „ 40(/3  q.  8,(3 / + 12J/2+ 5(2  1.3  .5  7 . 

a6-' : — - — : — „a*&c) 


204 8a' 4 (7 


the 


•2 .4. 6. 8 

Which,  when  a = 0,  or  e=  I,  becomes  only 

same  as  in  the  last  problem,  as  it  ought. 

Taking  here  the  same  numbers  as  in  the  last  problem, 
viz  l = |,  a = i,  uv=  2,  w = 28800,  g = 16TV  ; then 

\P\/~  ^ " = ’0040514,  and  the  series  is  1 + 006762  — . 

2wg 

•000175+  -000003,  &c.  = 1 006590  ; therefore  -0040514  X 

1-006590  = -0040965  =*  1 , is  the  time  of  one  whole  vi- 

24  + 

bration,  and  consequently  245}  vibrations  are  performed  in 
a second  ; which  were  250  in  the  last  problem. 


PROBLEM  49. 

It  is  proposed  to  determine  the  Velocity,  and  the  time  of  Vibra- 
tion of  a Fluid  in  the  arms  of  a Canal  or  bent  Tube. 

Let  the  tube  abcdef  have  its  two 
branches  ac,  ge  vertical,  and  the  lower 
part  cde  in  any  position  whatever,  the 
whole  being  of  an  uniform  diameter  or 
width  throughout.  Let  water,  or  quick- 
silver, or  any  other  fluid,  be  poured  in, 
till  it  stand  in  equilibrio,  at  any  hori- 
zontal line  bf.  Then  lei  one  surface  be  pressed  or  pushed 
down  by  shaking,  from  b to  c,  and  the  other  will  ascend 
through  the.  equal  space  fg  ; after  which  let  them  be  per- 
mitted 


PROMISCUOUS  EXERCISES. 


543 


Blitted  freely  to  return  The  surfaces  will  then  continually 
vibrate  in  equal  times  between  ac  and  eg.  The  velocity  and 
lines  of  which  oscillations  are  therefore  required 

When  the  surfaces  are  any  where  out  of  a horizontal  line, 
as  at  p and  the  parts  of  the  fluid  in  qdr,  on  each  side, 
below  du,  will  balance  each  other  ; and  the  weight  of  the 
part  in  pr,  which  is  equal  to  2ff,  gives  motion  to  the  whole. 
So  that  the  weight  of  the  part  2pf  is  the  motive  force  by 

which  the  whole  fluid  is  urged,  and  therefore  wt~  of  2f_f  the 
° wlirile  wt. 

accelerative  force.  Which  weights  being  proportional  to 
their  lengths,  if  l be  the  length  of  the  whole  fluid,  or  axis  of 

the  tube  filled,  and  a = fg  or  bc  ; then  is  the  accelerative 

force.  Putting  theref.  x = gp  any  variable  distance,  v the 

velocity,  and  t the  time  ; then  pf  = a — x,  and  — = / 

L 

• • • 4e  , . 

the  accelerative  force  ; hence  w or  2 g/s  = — (ax  — xx)  ; 

the  fluent  of  which  give  v2  = ~ (2ax  — x2),  and  v <= 

% / (4 g X ^ax~JL ) i3  the  general  expression  for  the  velocity 


at  any  term.  And  when  x = a,  it  becomes  v = 2a  y/  — for 
the  greatest  velocity  at  b and  f. 

Again,  for  the  time,  we  have  / or  — = + +/  — X 

the  fluents  of  which  give  t — X arc  to  versed  sine  — 


and  radius  1,  the  general  expression  for  the  time  And 
when  x = a,  it  becomes  t = \py/  — for  the  time  of  moving 


from  g to  f,  p being  = 3 1416,  and  consequently  \p^/  — 

the  time  of  a whole  vibration  from  g to  e,  or  from  c to  a. 
And  which  therefore  is  the  same,  whatever  ab  is,  the  whole 
length  l remaining  the  same. 

And  the  time  of  vibration  is  also  equal  to  the  time  of  the 
vibration  of  a pendulum  whose  length  is  \l,  or  half  the  length 
of  the  axis  of  the  fluid.  So  that,  if  the  length  l be  78£  inches, 
it  will  oscillate  in  1 second. 


Scholium.  This  reciprocation  of  the  water  in  the  canal,  is 
nearly  similar  to  the  motion  of  the  waves  of  the  sea.  For 
the  time  of  vibration  is  the  same,  however  short  the  branches 
are  provided  the  whole  length  be  the  same.  So  that  when 

the 


544 


PROMISCUOUS  EXERCISES. 


' I 

the  height  is  small,  in  proportion  to  the  length  of  the  canal, 
the  motion  is  similar  to  that  of  a wave,  from  the  top  to  the 
bottom  or  hollow,  and  from  the  bottom  to  the  top  of  the 
next  wave  ; being  equal  to  two  vibrations  of  the  canal  ; the 
whole  length  of  a wave,  from  top  top,  being  doable  the 
length  of  the  canal.  Hence  the  wave  will  move  forward  by 
a space  nearly  equal  to  its  breadth,  in  the  time  of  two  vibra- 
tions of  a pendulum  whose  length  is  (}l)  half  the  length  of 
the  canal,  or  one-fourth  of  the  breadth  of  a wave,  or  in  the 
time  of  one  vibration  of  a pendulum  whose  length  is  the 
whole  breadth  of  the  wave,  since  the  times  of  vibration  are 
as  the  square  rootAtof  their  lengths.  Consequently,  waves 
whose  breadth  is  eqs’al  to  39|  inches,  or  3§|  feet,  will  move 
over  3§f  feet  in  a sefcond,  or  195f  feet  in  a minute,  or  nearly 
2 miles  and  a quarteran  an  hour.  And  the  velocity  of  greater 
or  less  waves  will  be  increased  or  diminished  in  the  subdupli- 
oate  ratio  of  their  bnSulths 

Thus,  for  instance, &for  a wave  of  18  inches  breadth,  as 
^39}  • 39}  ::  yT8  T ^(39}  X 1 8)  =f  ^ 313=  26-5377 
the  velocity  of  the  wave  of  18  inches  breadth. 


E 


PROBLEM  50 

To  determine  the  Time  of  emptying  any  Ditch , or  Inundation, 
A'C.  by  a Cut  or  Notch,  from  the  Top  to  the  Bottom  of  it 

Let  x = ab  the  variable  height  of  water  at 

any  time  ; g 

b = ac  the  breadth  of  the  cut  ; 
d = the  whole  or  firsl  depth  of  water  ; 
a = the  area  of  the  surface  of  the  water 
in  the  ditch  ; 
g =?==■  16tV  feet 

The  velocity  at  any  point  n,  is  as  bd,  that  is,  as  the  ordi- 
nate de  of  a parabola  bec,  whose  base  is  ac,  and  altitude  ab. 
Therefore  the  velocities  at  all  the  points  in  ab,  are  as  all  the 
ordinates  of  the  parabola.  Consequently  the  quantity  of 
water  running  through  the  cut  abgc,  in  any  time,  is  to  the 
quantity  which  would  run  through  an  equal  aperture  placed 
all  at  the  bottom  in  the  same  time,  as  the  area  of  the  para- 
bola abc,  to  the  area  of  the  parallelogram  abgc,  that  is,  as 
2 to  3. 

But  y/ g : ^/x  : : 2 g : 2 y/gx  the  velocity  at  ac  ; therefore 
| X ty/gx  X bx  = }bx  gx  is  the  quantity  discharged  per 


second  through  abgc;  and  consequently  is  tje  ve- 

locity per  second  of  the  descending  surface  Hence  then 

= i the  fluxion  of  the  time  of 
Now 


4 bxjgx 


S a 

descending. 


: — x : : 1 


4 bx^gx 


PROMISCUOUS  EXERCISES. 


545 


Now  when  a the  surface  of  the  water  is  constant,  or  the 
ditch  is  equally  broad  throughout,  the  correct  fluent  of  this 

fluxion  gives  t = ,^A  X — for  the  general  time  of 

b 2 b</g  s/ilx  6 

sinking  the  surface  to  any  depth  x.  And  when  x = 0,  this 
expression  is  infinite  ; which  shows  that  the  time  of  a com- 
plete exhaustion  is  infinite. 

But  if  d = 9 feet,  b = 2 feet,  a = 21  X 1000  = 21000, 
and  it  be  required  to  exhaust  the  water  down  to  T'g  of  a foot 
deep  ; then  x = TV,  and  the  above  expression  becomes 

3X21000  3— i . . . , , . 

— - — - ^ — *4400  > or  Just  4 hours  for  that  time. 

And  if  it  be  required  to  depress  it  8 feet  or  till  1 foot  depth 
of  water  remain  in  the  ditch,  the  time  of  sinking  the  water 
to  that  point  will  be  43'  38". 

Again,  if  the  ditch  be  the  same  depth  and  length  as  before, 
but  20  feet  broad  bottom,  and  22  at  top  ; then  the  descend- 
ing surface  will  be  a variable  quantity,  and,  (by  prob.  16  Prac. 

Ex.  on  F orces) , it  will  be  - X 20000  ; hence  in  this  case  the 


the 

for 


ex- 


o r tu  i-  —Ax  , —500  90-f-x  . 

flux,  of  the  time,  or— , becomes  — - X — — x ; 

4 bx\/gx  ob^/ff  xv  x ’ 

. a . ‘r  , • , • . 1000  w .90  — x 90  — d. 

correct  fluent  of  which  is  t x ( ) 

3b  ^/g  v s/x  \/d  ' 

the  time  of  sinking  the  water  to  any  depth  x. 

Now  when  x — 0,  this  expression  for  the  complete 
haustion  becomes  infinite. 

But  if  . . x = 1 foot,  the  time  t is  42'  56"^. 

And  when  x — TV  foot,  the  time  is  3h  50'  28"$. 

PROBLEM  51. 

To  determine  the  Time  of  filling  the  Ditches  of  a Fortification 
6 Feet  deep  with  water,  through  the  Sluice  of  a Trunk  of  3 Feet 
Square,  the  Bottom  of  which  is  level  with  the  Bottom  of  the  Ditch , 
and  the  Height  of  the  supplying  Water  is  9 Feet  above  the  Bot- 
tom of  the  Ditch. 

Let  acdb  represent  the  area  of  the  vertical  sluice,  being 
a square  of  9 square  feet,  and  ab  level  with  the  bottom  of  the 
ditch.  And  suppose  the  ditch  filled  to  any  height  ae,  the  sur- 
face being  then  at  ef. 

Put  a — 9 the  height  of  the  head  or  supply  ; 

b = 3 = ab  = ac  ; 

g = 16  tV  ; 

a = the  area  of  a horizontal  section  of 
the  ditches  ; 

x — a — ae,  the  height  of  the  head 
above  ef. 


G 

C 

E 

A" 


Von.  IT. 


70 


B 

Then 


■546' 


PROMISCUOUS  EXERCISES. 


Then  y/ g : y/x  : : 2 g : 2^/gx  the  velocity  with  which  the 
water  presses  through  the  part  aefb  ; and  theref.  ‘dy/gx  X 
aefb  = 26  y/gx(a — x)  is  the  quantify  per  second  running 
through  aefb.  Also,  the  quantity  running  per  second  through 
ecdf  is  2 y/gx  X ecpf  = y b gx  (6  — a + x)  nearly. 
For  the  real  quantity  is,  by  proceeding  as  in  the  last  prob. 
the  difference  between  two  parab.  segs  the  alt.  of  the  one 
being  x , its  base  b,  and  the  alt  of  the  other  a — b ; and  the 
medium  of  that  dif  between  its  greatest  state  at  ab,  where  it 
is  t9„ad,  and  its  least  state  at  cd,  where  it  is  0,  is  nearl)  -{4ed. 
Consequently  the  sum  of  the  two,  or  }by/gx  (a  + 1 'b  — x) 
is  the  quantity  per  second  running  in  by  the  whole  sluice 

Hence  then  }b  y/  gx  X T = v is  the  rate 


ACDB. 

or  velocity  per  second 
ditches  : and  so  i)  : — a 


with 


A 

which 


1"  : i = 


the  water 
x — 6a 


rises  ik 

xcTi 

hy/g  i - X 


the  fluxion  of  the  time  of  filling  to  any  height  ae,  putting 
e — a -f-  1 1 6. 

Now  when  the  ditches  are  of  equal  width  throughout,  a 
is  a constant  quantity,  and  in  that  case  the  correct  fluent  of 

this  fluxion  is  t = ~--A  Xlog.  X^-  — ) the  ge- 

by/gc  y/L-y/a  y/ C + y/ 1 / 

neral  expression  for  the  time  of  filling  to  any  height  ae  or 
a — x,  not  exceeding  the  height  ac  of  the  sluice.  And 

when  x ==■  a c = a — b — d suppose,  then  t — - JA  X log. 

. by/gC 

is  the  time  of  filling  to  co  the  top  of 
the  sluice.  ✓ 

Again,  for  filling  to  any  height  gh  above  the  sluice,  x de- 
noting as  before  a — ag  the  height  of  the  head  above  gh, 
V gx  will  be  the  velocity  of  the  water  through  the  whole 
sluice  ad  : and  therefore  2 b2  y/  gx  the  quantity  per  second, 

and  v the  rise  per  second  of  the  water  in  the  ditches  ; 


• - = ^t3—  x — «»« 

V 20Jy/g  y/ X 

correct  fluent  of  which 


consequently  v : — x : : 1 ' : t — — 
general  fluxion  of  the  time  ; the 
being  0 when  x = a — b — d,  is  t = , — — (y/  d - y/x)  the 

b-y/SK 

time  of  filling  from  cn  to  gh. 

Then  the  sum  of  the  two  times,  namely,  that  of  filling 
from  ab  to  cd,  and  that  of  filling  from  cd  to  gh,  is 


by/g 


y/tl--  y/X  | 6 . . y/r-l-  y/ a y/  — yf  , 

^ b y/c  ° ^y/C—y/a  ’ y/C-\-y/d' 


■)] 


for  the  whole 
time 


4 


PROMISCUOUS  EXERCISES. 


547 


time  required 

this  becomes  , A—  [ 
V? 

= 0 035772  7 7a, 
length  and 


And,  using  the  numbers 

%/•&—  v/  > , 6 


X 1 ( 


the  prr>b„ 


4g  ^ s/ Lt  2 — j «/' 42-1-  j ' 

the  time  in  terms  ot  a the  area  of  the 
breadth,  or  horizontal  section  of  the  ditches. 
And  if  we  suppose  that  area  to  be  200t)00  square  feet,  the 
lime  required  will  be  7154  ',  or  lh  59'  14'' 

And  if  the  sides  of  the  ditch  slope  a little,  so  as  to  be  a 
little  narrower  at  the  bottom  than  at  top,  the  process  will  be 
nearly  the  same,  substituting  for  a its  variable  value,  as  in 
the  preceding  problem  And  the  time  of  tilling  will  be  very 
nearly  the  same  as  that  above  determined. 

PROBLEM  25. 

But  if  the  Water , from  which  the  Ditches  are  to  be  filled , 
be  the  Tide , which  at  how  Water  is  below  the  Bottom  of  the 
Trunk,  and  rises  to  9 Feet  above  the  Bottom  of  it  by  a regular 
Rise  of  One  Foot  in  Half  an  H >ur  ; it  is  required  to  ascertain 
the  Time  of  Filling  it  to  6 Feet  high , as  before  in  the  last 
Problem. 

Let  a-’db  represent  the  sluice  ; and  when  the  tide  has  risen 
Jo  any  height  gh,  below  cd  the  top  of  the  sluice,  without  the 
ditches,  let  ef  be  the  mean  height  of  the  water  within, 
And  put  b — 3 = ab  = ac  ; * 

g = ; i S 

a = horizontal  section  of  the  ditches  ; 
x = ag  ; 

2 = AE. 

Then  v/ ’g  : ^/eg  : : 2 g : 2 ^/g  (x — z ) the  velo- 
city of  the  water  through  aefb  ; and 
y/K  : t/EG  : : I?  : iv/  o(x — z)  mean  vel.  through  eghf  : 
theref.  2bz  f g(x — z)  is  the  quantity  per  sec.  through  aefb  ; 
and  1 5(x  — 2)  ^/^(x — ■?)  is  the  same  through  eghf  ; 
conseq  %b^/  g X (2r  + z)  (x — z)  is  the  whole  through 
aghb  per  second.  This  quantity  divided  by  the  surface  a, 

2b  a/p* 

gives  * (2r  z)  — zl  ~ v vei°«>ty  per  second 

with  which  ef,  -or  the  surface  of  the  water  in  the  ditches, 
rises.  Therefore 

......  M'  oA  v = 

V : Z • • 1 • t X ; ; . 

V 2b  ^g  ( x -rs)s(x— 2) 

But,  as  gh  rises  uniformly  i foot  in  30  or  1800 ' there- 
fore 1 : ag  : : 1800  : l<300r  = t the  time  of  the  tide  rising 

3 a. 

through  ag  ; conseq.  i — 1800x  = — — X 

2 b</g  (2x-fa)V(*--z), 
ormz  =(2x-}-2-)v/(x — z)  . xis  the  fluxional  equa.  expressing 

the  relation  between  x and  z ; where  m = — = 

12004  ^5"  231 

er  13iff  when  a — 200000  square  feet.  Now 


D 

H 

F 

TB 


548 


PROMISCUOUS  EXERCISES. 


Now  to  find  the  fluent  of  this  equation,  assume  2 ■== 

_5  ,8  l_l  13 

Aa:2  + Ba:2  + ca!  * +dx  2 &c.  So  shall 

1 0 


\ 1 a A a2  4.4b  1 a34-4ab4-8c 

S(*-z)  = X*--X * TV 


2 8"  16 
A 3.  1J 

2a:  + 2 = 2x  -J-  ax24"Ea:2+cx  2 &c. 
(2x+2)x/(x-2)i=2a:^i#-^  x2i  A +6ab  ~2 


&c. 


x 2 X &ZC. 


and  mz  = |mAX2^--(-|OTBx2.r+  V mcx2x  -{-  V4  mDX  2 a:  &c. 
Then  equate  the  coefficients  of  the  like  terms, 


so  shall 

■f-mA  = 2, 

§WIB  = 0, 

V me  = — }a2 

J24/nD  = 

&c.  ; 


and  consequently 
4 

5m’ 

B = 0, 

■ — _ 24 
C _ 275m  3 5 

16 

° 875m~’ 

&C. 


Which  values  of  a,  b,  c,  &c.  substituted  in  the  assumed 
value  of  2,  give 

4 i-  24  V 16 


5m  * 275m  3 875m  4 

4 5 

or  2 = —re3  very  nearly. 


&c. ; 


And  when  a;  = 3 = ac,  then  2 = -886  of  a foot,  or  10| 
inches,  = ae,  the  height  of  the  water  in  the  ditches  when 
the  tide  is  at  cd  or  3 feet  high  without,  or  in  the  first  hour  anfl 
half  of  time. 

Again,  to  find  the  time,  after  the  above,  when 
ef  arrives  at  cd,  or  when  the  water  in  the 
ditches  arrives  as  high  as  the  top  of  the  sluice. 

The  notation  remaining  as  before, 
then  Ibz^/gioc — 2)  per  sec.  runs  through  af, 
and  56(3 — z)\/g (-a? — 2)  per  sec.  thro’  ed  nearly; 
therefore  |6v/gX(12  + 2)  (.r  — 2)  is  the  whole  per  second 

through  ad  nearly. 

conseq.  ——  X (12  + 2)  (z  — 2)  = v is  the  velocity  per 


I 

Gh~ 

Ch- 

E -- 


B 


5a 

second  of  the  point  e ; and  therefore, 

* 3A  v/  - — 

z : : 1 • t , .v  . X 


2 b-Sg 


f»i=  (12+2)v/(a:  — 2)  . i,  where  m 


(12  T-2)v'(X-2) 

A 


1800  ic,  or 


720b 


23  a%  nearly. 
Assume 


PROMISCUOUS  EXERCISES. 


549 


3.  4.  5.  6_ 

Assume  z = ajt2+bx2-|-cx2+dx2  &c. 


V (x—z)  = x*-  ^x-  - 


a a a2  -|-4b  1 


8 


So  shall 

A2-f  4AB+  8C_|8t 
16 


x2&c  ; 


12 -{-z  — 12  -f-  A.t2 -[-BX2+CX2  &c  ; 

(12+z)  . y/(x — 2)  . i=12a?^jr— 6ax^x — (f  a2  + 6b)x  2,r  &c  ; 

. 1 . 3..  .a, 

mz  --  %m\x*x  Jr%,rn^x2 x-\-^mcx2x  &c. 

Then,  equating  the  like  terms,  &c.  we  have 

d = 


8 


A = — , B 
m 


24 


96 


64 


m2  ’ 5m s’  3m4 

„ ' 8 a 24  „ , 96  4 , 64 

Hence  2 = — x2 x2  4-; x2+^ — — x 

m m2  07n  3 

8 3 

Or  2 = — x^nearlv. 

7K  J 


3»I4 


nearly,  &c. 
&c. 


But,  by  the  first  process,  when  x = 3,  2 = ’886  ; which 
substituted  for  them,  we  have  2 = -886,  and  the  series  = 
1‘63  : therefore  the  correct  fluents  are 

8 2 24 

2 — -886  = - 1-63  + -x2  — —x2  &c. 

m m2 

8 2 24 

or  2 + "774  - — x2 — x2  &c. 

m m2 

And  when  2=3=  ac,  it  gives  x = 6 ‘369  for  the  height 
of  the  tide  without,  when  the  ditches  are  filled  to  the  top  of 
the  sluice,  or  3 feet  high  ; which  answers  to  3h  11'  4". 

Lastly,  to  find  the  time  of  rising  the  remaining  3 feet  above 
the  top  of  the  sluice  ; let 

x = cg  the  height  of  the  tide  above  cd,  Gy iH 

2 = ce  ditto  in  the  ditches  above  cd  ; Ei- iE 

and  the  other  dimensions  as  before.  C D 

Then  y/  g : y/  es  : : 2g  : 2 y/  g (x  — 2)  = the 
velocity  with  which  the  water  runs  through  the 


whole  sluice  ad  ; conseq.  adX2  y/  g(x — 2)  = A 


B 


18  y/  g(x  — 2)  is  the  quantity  per  second  running  through  the 
sluice,  and  (x  — z)  — v the  velocity  of  2,  or  the  rise 

of  the  water  in  the  ditches,  per  second  ; hence  v : z : : ly'  : } 

2)#is  the 


: — — X - = 1800^,  and  mz  —Xy/{x- 

™y/g  v/U  -2)  v v 

a 3200 


flusional  equation  ; where  m 


1802  y/  g 2079’ 


* Note . The  fluxional  equation  mi 
without  series. — Editor, 


may  be  integrated 

To 


PROMISCUOUS  EXERCISES. 


^50 


■x*x 


To  find  the  fluent, 

a.  , 4 5.  6. 

Assume  z — ax2  -j-Bx^-f-cx2  -}-dx2  &c. 

Then  x — z — jt-ax^-bx^ — cx^  &c. 

• ./  \ A • A \ . A-  'J-4b  g- 

jx2x 8 

1 . 3 . 3 . 

fflz  = ■|«AX,jr+  |«Bx2x+|nrx?x  &c. 
Then  equating  ffle  like  terms  gives 

_ 2 — ! 1 - 1 . 

A — - — , B — , C — , D= . &C. 

3/i  67j2  9 On*  810/.  4 

„ 2 3 1 1 5.  1 

Hence  z = — x2 x2  q x2 

3-<  6*2  T90«a  810«4 


r be. 


t3  &c. 


But,  by  the  second  case,  when  z — 0,  x = 3-369,  which 
being  used  in  the  series,  it  is  1 936  ; therefore  the  correct 

2 3 i 

fluent  is  z — — 1 936  q-  — x2  — - — x-  8zc.  And  when 

3 n 6 n* 

z = 3,  x — 7 ; the  heights  above  the  top  of  the  sluice  ; 
answering  to  6 and  10  feet  above  the  bottom  of  the  ditches. 
ri  hat  is,  for  the  water  to  rise  to  the  height  of  6 feet  within 
the  ditches,  it  is  necessary  for  the  tide  to  rise  to  10  feet  with- 
out, which  just  answers  to  5 hours  ; and  so  long  it  would 
take  to  fill  the  ditches  6 feet  deep  with  water,  their  horizontal 
area  being  200000  square  feet. 

Further,  when  x = 6,  then  z — 2 117  the  height  above 
the  top  of  the  sluice  ; to  which  add  3,  the  height  of  the  sluice, 
and  the  sum  5-117,  is  the  depth  of  water  in  the  ditches  in 
4 hours  and  a half,  or  when  the  tide  has  risen  to  the  height  of 
9 feet  without  the  ditches. 

Note.  In  the  foregoing  problems,  concerning  the  efflux 
of  water,  it  is  taken  for  granted  that  the  velocity  is  the  same 
as  that  which  is  due  to  the  whole  height  of  the  surface  of  the 
supplying  water  : a supposition  which  agrees  with  the  prin- 
ciples of  the  greater  number  of  authors  : though  some  make 
the  velocity  to  be  that  which  is  due  to  the  half  height  only  : 
and  others  make  it  still  less. 

Also  in  some  places,  where  the  difference  between  two 
parabolic  segments  was  to  be  taken,  in  estimating  the  mean 
velocity  of  the  water  through  a variable  orifice,  1 have  used 
a near  mean  value  of  the  expression  ; which  makes  the  ope- 
ration of  finding  the  fluents  much  more  easy,  and  is  at  the 
same  time  sufficiently  exact  for  the  purpose  in  hand. 

We  may  further  add  a remark  here  concerning  the  method 
of  findiug  the  fluents  of  the  three  fluxional  forms  that  occur 
in  the  solution  of  this  problem,  viz  the  three  forms  in'z  = 
(2x  -f  z)  (x  — z)x,  and  mz  = (12  + z)x/{x— z)x,  and 


PROMISCUOUS  EXERCISES. 


551 


fjijt  = y/  (x  — z)x  the  fluents  of  which  are  found  by  assum- 
ing the  fluent  mz  in  an  infinite  series  ascending  in  terms  of 
x witii  indeterminate  coefficients  a,  b,  c,  &c  which  coeffi- 
cients are  afterwards  determined  in  the  usual  way,  by  equat- 
ing the  corresp  nding  terms  of  two  similar  and  equal  series, 
the  one  series  denoting  one  side  of  the  fluxional  equation,  and 
the  other  series  the  other  side  By  similar  series,  is  meant 
when  they  have  equal  or  like  exponents  ; though  it  is  not 
necessary  that  the  exponents  of  all  the  terms  should  he  like 
or  pairs,  but  only  some  of  them,  as  those  that  are  not  in  pairs 
will  be  cancelled  or  expelled  by  making  their  coefficients  = 
0 or  nothing.  Now  the  general  way  to  make  the  two  series 
simitar,  is  to  assume  the  fluent  z equal  to  a series  in  terms  of 
x,  either  ascending  or  descending,  -as  here 

z = xr  + xT*‘  + xrYls  Szc  for  ascending, 
or  z = xT  + xr~3  + xr~Zs  Szc.  for  a descending 
series,  having  the  exponents  r,  r ± s,  r ± 2s,  Szc  in  arith- 
metical progression,  the  first  term  r,  and  common  difference 
s ; without  the  general  coefficients  a,  b,  c,  &c  till  the  values 
of  the  exponents  be  determined.  In  terms  of  this  assumed 
series  for  z,  find  the  values  of  the  two  sides  of  the  given  flux- 
ional equation,  by  substituting  in  it  the  said  series  instead  of 
z ; then  put  the  exponent  of  the  first  term  of  the  one  side 
equal  that  of  the  other,  which  will  give  the  value  of  the  first 
exponent  r ; in  like  manner  put  the  exponents  of  the  two  2d 
terms  equal,  which  will  give  the  value  of  the  common  differ- 
ence s ; and  hence  the  whole  series  of  exponents  r,  r ± s,  r 
± 2s  Szc.  becomes  known. 

Thus,  for  the  last  of  the  three  fluxional  equations  above 
mentioned,  viz.  mz  ~ y/  {x—z)x,  or  only  z — y/  {x  — zx)  ; 
having  assumed  as  above  z = xT  + xr**  Szc.  and  taking  the 
fluxion,  then  z = xr~l  x + xr+'t-1  x + &c.  omitting  the 
coefficients  ; and  the  other  side  of  (he  equation  y/  (x—z)x~ 

y/  (x  — xr — xr<'s  Szc)  = x^x— xr~^x  Szc.  Now  the  expo- 
nents of  the  first  terms  made  equal,  give  r — 1 = theref. 
t — 1 -f-  i = f ; and  those  of  the  2d  terms  made  equal, 
give  r+s — 1 =r- — theref  s — 1 •-  — and  s = 1 — £ = J; 
conseq.  the  whole  assumed  series  of  exponents  r,  r + 
r -f-  2s,  &c.  become  §,  f , f , Szc.  as  assumed  above. 

Again,  for  the  2d  equation  mz  or  z — (12+2')  + (re  — 2)i? 
= (a+z)  y/  (x  — z)x  ; assuming  2 = xr  + xr's  Szc  as  before, 

then  z = xr~l  x +xrtl—+  Szc  and  y/  {x—  z)x  = x^x  — xr~^'x 
Szc.  both  as  above  ; this  mult,  by  a + z or  a + x,r+arr,‘J  &c. 

gives  ax  $ x — axr~hx  Szc  : then  equating  the  first  exponents 
gives  r— l = i or  r = f ,and  r + s — l=r—  i,  or  s=l  — i=i  ; 

* hence 


552 


PROMISCUOUS  EXERCISES. 


hence  the  series  of  exponents  is  f , f , f , &c.  the  same  as  the 
former,  and  as  assumed  above. 

Lastly,  assuming  the  same  form  of  series  for  ^ and  2 as  in 
the  above  two  cases,  for  the  1st  fluxional  equation  also,  viz. 

mz  =(2 x-\-z)x/(x—z)x  : then  '/(x— z)x—x^x  — xr~7x  &c. 

which  mult,  by  2x  + z,  gives  2x%x  — xr**x  &c.  : here  equat- 
ing the  first  exponents  gives  r — 1 — f or  r = ■§,  and  equat- 
ing the  2d  exponents  gives  r -f-  3 — 1 = r + 4,  or  s = s j 
hence  the  series  of  exponents  in  this  case  is  f , f , y , &c.  as 
used  for  this  case  above.  Then,  in  every  case  the  general 
coefficients  a,  b,  c,  &c.  are  joined  to  the  assumed  terms  xr, 
xr*s,  &c.  and  the  whole  process  conducted  as  in  the  three 
series  just  referred  to. 

Such  then  is  the  regular  and  legitimate  way  of  proceeding, 
to  obtain  the  form  of  the  series  with  respect  to  the  expo- 
nents of  the  terms.  But.  in  many  cases  we  may  perceive  at 
sight,  without  that  formal  process,  what  the  law  of  the  ex- 
ponents will  be,  as  I indeed  did  in  the  solutions  in  the  series 
above  referred  to  ; and  any  person  with  a little  practice  may 
easily  do  the  same.  | 

PROBLEM  53. 

To  determine  the  fall  of  the  Water  in  the  Arches  of  a Bridge. 

The  effects  of  obstacles  placed  in  a current  of  water,  such 
as  the  piers  of  a bridge,  are,  a sudden  steep  descent,  and  an 
increase  of  velocity  in  the  stream  of  water,  just  under  the 
arches,  more  or  less  in  proportion  to  the  quantity  of  the  ob- 
struction and  velocity  of  the  current  ; being  very  small  and 
hardly  perceptible  where  the  arches  are  large  and  the  piers 
few  or  small,  but  in  a high  and  extraordinary  degree  at 
London-bridge,  and  some  others,  where  the  piers  and  the 
sterlings  are  so  very  large,  in  proportion  to  the  arches.  This 
is  the  case,  not  only  in  such  streams  as  run  always  the  same 
way,  but  in  tide  rivers  also,  both  upward  and  downward,  but 
much  less  in  the  former  than  in  the  latter.  During  the  time 
of  flood,  when  the  tide  is  flowing  upward,  the  rise  of  the 
w'ater  is  against  the  under  side  of  the  piers  ; but  the  differ- 
ence between  the  two  sides  gradually  diminishes  as  the  tide 
flows  less  rapidly  towards  the  conclusion  of  the  flood.  When 
this  has  attained  its  full  height,  and  there  is  no  longer  any 
Current,  but  a stillness  prevails  in  the  water  for  a short  time, 
the  surface  assumes  an  equal  level,  both  above  and  below 
bridge  But  as  soon  as  the  tide  begins  to  ebb  or  return 
again,  the  resistance  of  the  piers  against  the  stream,  and  the 
contraction  of  the  waterway,  cause  a rise  of  the  surface  above 
and  under  the  arches,  with  a full  and  a more  rapid  descent  in 


PROMISCUOUS  EXERCISES. 


553 


the  contracted  stream  just  below.  The  quantity  of  this  rise, 
and  of  the  consequent  velocity  below,  keep  both  gradually 
increasing,  as  the  tide  continues  ebbing,  till  at  quite  low 
water,  when  tbe  stream  or  natural  current  being  the  quick- 
est, the  fall  under  the  arches  is  the  greatest.  And  it  is  the 
quantity  of  this  fall  which  it  is  the  object  of  this  problem  to 
determine. 

Now,  the  motion  of  free  running  water  is  the  consequence 
of,  and  produced  by  the  force  of  gravity,  as  well  as  that  of  any 
other  falling  body.  Hence  tbe  height  due  to  the  velocity, 
that  is,  the  height  to  be  freely  fallen  by  any  body  to  acquire  the 
observed  velocity  of  the  natural  stream,  in  the  river  a little 
way  above  bridge,  becomes  known.  From  the  same  velocity 
also  will  be  found  that  of  the  increased  current  in  the  narrow- 
ed way  of  the  arches,  by  taking  it  in  the  reciprocal  proportion 
of  the  breadth  of  the  river  above,  to  the  contracted  way  in  the 
arches  ; viz.  by  saying,  as  the  latter  is  to  the  former,  so  is  the 
first  velocity,  or  slower  motion,  to  the  quicker.  Next,  from 
this  last  velocity,  will  be  found  the  height  due  to  it  as  before, 
that  is,  the  height  to  be  freely  fallen  through  by  gravity,  to 
produce  it.  Then  the  difference  of  these* two  heights,  thus 
freely  fallen  by  gravity,  to  produce  the  two  velocities,  is  the 
required  quantity  of  the  waterfall  in  the  arches  ; allowing 
however,  in  the  calculation  for  the  contraction,  in  the  narrow- 
ed passage,  at  the  rate  as  observed  by  Sir  I.  Newton,  in  prop. 
36  of  the  2d  book  of  the  Principia,  or  by  other  authors,  being 
nearly  in  the  ratio  of  25  to  21.  Such  then  are  the  elements 
and  principles  on  which  the  solution  of  the  problem  is  easily 
made  out  as  follows. 


Let  b = the  breadth  of  the  channel  in  feet ; 

v = mean  velocity  of  the  water  in  feet  per  second  ; 
c = breadth  of  the  waterway  between  the  obstacles. 

21 

Now  25  : 21  : : c : — c,  the  waterway  contracted  as  above. 

21  256 

And  — c : b : : v v,  the  velocity  in  the  contracted  way. 

25  21c 

Also  32-  : -c-  : : 16  : g^v2,  height  fallen  to  gain  the  velocity  z\ 

And322  :(^y)2  : : 16  : (~)2  X ^o2 , ditto  for  the  vel.^‘7;. 
21c  ' 21c.  21c 

Then  C— ')2  X— — is  the  measure  of  the  fall  required. 

^21c  64  64 

Or[(“^)2  - l]X-^is  a rule  for  computing  the  fall. 

1 *4^62  - c2  _ 

Or  rather  — 1 Xt,2  very  nearly,  for  the  fall. 

£4c2 


Vol.  U. 


Ex^m- 


554 


PROMISCUOUS  EXERCISES. 


Exam.  1.  For  London-bridge. 

By  the  observations  made  by  Mr.  Labelye  in  1746, 

The  br^adtlj  of  the  Thames  at  London-bridge  is  926  feet  ; 
The  sum  of  the  waterways  at  the  time  of  low-water  is  236  ft.; 
Mean  velocity  of  the  stream  just  above  bridge  is  3|  ft.  per  sec. 
But  under  almost  all  the  arches  are  driven  into  the  bed  great 
numbers  of  what  are  called  dripshot  piles,  to  prevent  the  bed 
from  being  washed  away  by  the  fall.  These  dripshot  piles 
still  further  contract  the  waterways,  at  least  J-  of  their  measur- 
ed breadth,  or  near  39  feet  in  the  whole  ; so  that  the  waterway 
will  be  reduced  to  197  feet,  or  in  round  numbers  suppose  200 
feet. 

Then  b = 926,  c = 200,  v — S±. 

1-4262  1217616—40000 

Hence  — = — — = *46, 


64c2 

192 


64X40000 


And  v2  =-—  = 10  3V. 

Theref.  46  X lO/g  =4-73  ft.=4  ft.  8f  in.  the  fall  required 
By  the  most  exact  observations  made  about  the  year  1736.. 
the  measure  of  tftfnall  was  4 feet  9 inches. 


Exam.  2.  For  Westminster-bridge. 


Though  the  breadth  of  the  river  at  Westminster-bridge  is 
1220  feet ; yet,  at  the  time  of  the  greatest  fall,  there  is  water 
through  only  the  13  large  arches,  which  amount  to  bat  820 
feet ; to  which  adding  the  breadth  of  the  12  intermediate 
piers,  equat  to  174  feet,  gives  994  for  the  breadth  of  the 
river  at  that  time  ; and  the  velocity  of  the  water  a little  above 
the  bridge,  from  many  experiments,  is  not  more  than  ft 
per  second.  . 

Here  then  b 

1-4262  _C2 


994,  c = 820,  v — 2i  = f . 


Hence 


And  v2 


64c2 

812 

16 


1403011-672400 
64  X 672400 


= -01722. 


= 5* 


Theref.  -01722  X 5Ty  = -0872  ft.=l  in.  the  fall  required  ; 
which  is  about  half  an  inch  more  than  the  greatest  fall  ob- 
served by  Mr.  Labelye. 

And,  for  Blackfriar’s-bridge,  the  fall  will  be  much  the  same 
jiS  that  of  Westminster. 


ADDITIONS- 


C 555  ] 


ADDITIONS, 


BY  THE  EDITOR,  R.  ADRAIN.  > 


New  method  of  determining  the  Angle  contained  by  the  chords 
of  two  sides  of  a Spherical  Triangle. 

See  prob.  v.  page  77,  vol.  2. 


THEOREM. 

If  any  two  sides  of  a Spherical  Triangle  be  produced  till  the 
continuation  of  each  side  be  half  the  supplement  of  that 
side,  the  arc  of  a great  Circle  joining  the  extremities  of  the 
sides  thus  produced  will  be  the  measure  of  the  Angle  con- 
tained  by  the  chords  of  those  two  sides. 


DEMONSTRATION. 


Let  the  two  sides  ab,  ac  of  the  spherical 
triangle  abc  be  produced  till  they  meet  in 
g,  and  let  the  supplements  bo,  cg,  be  bi- 
sected in  d and  e,  also  let  the  chords  ^wib, 
amc  of  the  arcs  ab,  ac  be  drawn  ; and 
the  great  circular  arc  de  will  be  the  mea- 
sure of  the  rectilineal  angle  contained  by  the 

Chords  A/71B,  AWC. 

Let  the  diameter  ag  be  the  common  section  of  the  planes 
of  abg,  acg,  and  f the  centre  of  the  sphere,  from  which  draw 
the  straight  lines  fd,  fe. 

Since,  by  hypothesis,  ge  is  the  half  of  gc,  therefore  the 
angle  at  the  centre  gfe  is  equal  to  the  angle  at  the  circumfe- 
rence gamc  (theo.  49.  Geom.),  and  therefore  awc  andFE,  being 
in  the  same  plane,  are  parallel  : in  like  manner,  it  is  shown 
that  fd  and  awib  are  parallel,  and  therefore  the  rectilineal 
angles  bac  and  dfe,  are  equal,  and  consequently,  since  de  is 
the  measure  of  the  angle  dfe,  it  is  also  the  measure  of  the 
angle  contained  by  the  chords  awb  and  A7ic.  q.  e.  d. 

New 


556 


ADDITIONS. 


New  * method  of  determining  the  oscillations  of  a Variable 
Pendulum . 

The  principles  adopted  by  Dr.  Hutton  in  the  solution  of 
his  45th  problem,  page  537,  vol  2,  are,  in  my  opinion, erroneous. 
He  supposes  the  number  cf  vibrations  made  in  a given  parti- 
cle of  time  to  depend  on  the  length  of  the  pendulum  only, 
without  considering  the  accelerative  tension  of  the  thread  ; so 
that  by  his  formula  we  have  a finite  number  of  ^ ibrations  per- 
formed in  a finite  time  by  the  descending  weight,  even  when 
the  ascending  weight  is  infinitely  small  or  nothing  Besides, 
the  stating  by  which  he  finds  the  fluxion  of  the  number  of  vi- 
brations, is  referred  to  no  geometrical  or  mechanical  princi- 
ple, and  appears  to  be  nothing  but  a mere  hypothesis.  The 
following  is  a specimen  of  the  method  by  which  sud.  prob- 
lems may  be  solved  according  to  acknowledged  principles. 

PROBLEM. 


If  two  unequal  weights  m and  m connected  by  a thread  pas- 
sing  jreely  over  a pully,  are  suspended  vertically,  and  exposed 
to  the  action  of  common  gravity,  it  is  required  to  investigate  the 
number  of  vibrations  made  in  a given  time  by  the  greater  weight 
m,  supposing  it  to  descend  from  the  point  of  suspension,  and  to 
make  indefinitely  small  removals  from  the  vertical. 


SOLUTION. 


Let  the  summit  a of  a vertical  abcde  be  the 
point  from  w hich  m descends,  b any  point  in  ae 
taken  as  the  beginning  of  the  plane  curve  Bmon 
described  by  m,  which  is  connected  with  m by 
the  thread  Am.  Let  me  be  at  right  angles  to  ae, 
and  put  ac  = x,  cm  = y,  a m = r ; also  let  r,  t 
and  t be  the  times  of  the  descent  of  m through 
the  vertical  spaces  ab,  ac,  and  bc  ; g = 32^ 
feet,  = !he  measure  of  accelerative  gravity ; 
f — the  measure  of  the  retarding  force  which 
the  tension  of  the  thread  exerts  on  m in  the  direction  «ia,  and 
c = the  indefinitely  small  horizontal  velocity  of  m at  b. 

/jf 

As  r : x : :f:—  — the  vertical  action  of  the  tension  on  m . 

r 

fx 

and  therel.  g ———  the  true  accelerative  force  with  which  in 

r 

is  urged  in  a vertical  direction. 

Again. 


ADDITIONS. 


557 


Again,  r : y : the  horizontal  action  on  produced 

by  the  tension  of  the  thread  a m.  Thus  the  whole  accelerative 
forces  by  which  m is  urged  in  directions  parallel  to  x and  y, 

f X fit  . 

are  g— — , and  — , the  former  of  these  forces  tending  to  in- 
crease x,  and  the  latter  to  diminish  y ; and  therefore  by  the 
general  and  well  known  theorem  of  variable  motions  (See 
Mec.  Cel.  B.  1,  Chap,  2),  we  have  the  two  equations 

.*  ^ fx  ^ fy 

i2  r 't-  r 


But  by  hypothesis,  the  angle  wac  is  indefinitely  small,  we  have 
therefore  --=  1,  and  f = — — = a given  quantity  ; our  first 

T -pn 

fluxional  equation  therefore  becomes 


•X  j* 

\ \2-S-l i 

of  which  the  proper  fluent  is  x = i (g  — J)  l2  : and  by  substitu- 
ting for  x the  value  just  found,  our  second  fluxional  equation 
becomes. 


v *f. 
t2  ff-f 


y 1 2 y 

- — or 

«2  t'J 


py  — 0,  (putting  p 


_ ~f 4i, 

g-f  »»- 


0- 


Now  when  p is  less  than  1,  let  q — y/  \ — p,  and  in  this  case 
the  correct  fluent  of  the  equation—,,-  -\-py— 0,  is  easily  found 
to  be 

from  which  equation  it  is  manifest  that  as  t increases  y also  in- 
creases, so  that  m never  returns  to  the  vertical,  and  there  are 
no  vibrations.  Again,  when  p — j , the  correct  fluent  of  the 
same  fluxional  equation  is 


r=V tT  • hJP-  Ios-  0- 


So  that  in  this  case  also,  when  t increases  y increases,  and  the 
body  in  never  returns  to  the  vertical.  Since  in  this  case  p = 

4 = I,  therefore  17m'  ==  m,  and  therefore  by  this  case 
m — m 

arid  the  preceding,  there  are  no  vibrations  performed  by  the 
descending  weight  m wrhen  it  is  equal  to  or  greater  thau 
17  times  the  ascending  weight  m. 

But 


558 


ADDITIONS. 


But  when/?  is  greater  than  a,  put  n = — a,  and  in  this 

case  the  correct  equation  of  the  fluents  is 

— . sin.  (n  . hyp.  log. -). 
c n v t' 

This  equation  shows  us  that  we  shall  hare  y — 0 as  often  as 
n.  hyp.  log.  - becomes  equal  to  any  complete  number  of  semi- 
circumferences : if  therefore  * = 3- 1416,  and  n = any  num- 
ber in  the  series  1,  2,  3,  4,  5,  &c.  we  can  have  y = 0 only 

when  n.  hvp.  log.  - = N?r,  from  which  we  have  — , 

J r r tz=r:cn 

supposing  hyp.  log.  e = 1,  and  therefore 

N5T 


which  shows  the  relation  between  the  number  of  vibrations  n 
and  the  time  t in  which  they  are  performed. 

Hence  it  is  manifest,  that  the  times  or  durations  of  the  seve- 
ral successive  vibrations  constitute  a series  in  geometrical 
progression. 


logarithms 


LOGARITHMS 


or  THE 

LUMBERS 

FROM 

1 to  1000. 


_ 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

j 

1 

0-000000 

26 

1-414973 

51 

1-707570 

76 

1-880814 

it 

0"301030 

27 

1-431364 

52 

T716003 

77 

1*88649 1 

3 

0-477121 

28 

1-447158 

53, 

T724276 

78 

1-892095 

4 

0'602060 

29 

1-462398 

54- 

1-732394 

79 

T897627 

5 

0-698970 

30 

1-477121 

55 

1-740363 

80 

1-903090 

6 

0-778151 

31 

1-491362 

T505150 

56 

57 

1-748188 

81 

1-908485 

7 

0-845098 

32 

1-755875 

82 

1-913814 

8 

0-903090 

33 

1-518514 

58 

1-763428 

83 

1*919078 

9 

0-954243 

34 

1-531479 

59 

1-770852 

84 

1-924279 

10 

1-000000 

35 

1-544068 

60 

1-778151 

85 

1-929419 

! li 

1-041393 

36 

1-556303 

61 

1-785330 

86 

1-934498 

12 

1-079181 

37 

1-568202 

62 

1-792392 

87 

1-939519 

13 

1-113943 

38 

1-579784 

63 

1-799341 

88 

1-944483 

14 

1-146128 

39 

1-591065 

64 

1-806180 

89 

1-949390 

15 

1-176091 

40 

1 602060 

65 

1-812913 

90 

1-954243 

16 

1-204120 

41 

1-612784 

66 

1-819544 

91 

1-959041 

17 

1-230449 

42 

1-623249 

67 

1-826075 

92 

1-963788 

| 18 

1-255273 

43 

1-633468 

68 

1-832509 

93 

1-968483 

19 

1-278754 

44 

1-643453 

69 

1-838849 

94 

1-973128 

20 

1-301030 

45 

1-653213 

70 

1-845098 

95 

1-977724 

21 

1-322219 

46 

1-662758 

71 

1-851258 

96 

1-982271 

22 

1-342423 

47 

1-672098 

72 

1-857333 

97 

1-986772 

23 

1-361728 

48 

1 681241 

73 

1 *863323 

98 

1-991226 

24 

1-380211 

49 

1-690196 

74 

1-869232 

99 

1-995635 

25 

1-397940 

50 

1 698970 

75 

1-875061 

100 

2-000000 

N-  B.  In  the  following  table,  in  the  last  nine  columns  of  each  page, 
where  the  first  or  leading  figures  change  from  9’s  to  0’s,  points  or 
dots  are  now  introduced  instead  of  the  0’s  through  the  rest  of  the 
line,  to  catch  the  eye,  and  to  indicate  that  from  thence  the  cor- 
responding natural  number  in  the  first  column  stands  in  the  next 
lower  line,  and  its  annexed  first  two  figures  of  the  Logarithm  in 
the  second  column. 


LOGARITHMS. 


I*- 

0 

1 

2 

3 

4 

5 

6 

1 7 

8 

9 

1 100 

000000 

0434 

0868 

1301 

1734 

2166 

2598  3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894  732 1 

7748 

8174 

102 

8600 

9026 

9451 

9876 

• 300 

• 724 

1147  1570 

1993 

2415 

1103 

012837 

3259 

3680 

4100 

4521 

4940 

5360  5779 

6197 

6616 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532:9947 

• 361 

* 77 5 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664'407  5 

4486 

4896 

.106 

5306 

5715 

6125 

6533 

6942 

7350 

7757:8164 

8571 

897-8 

j 107 

9384 

9789 

• 195 

• 600 

1004 

1408 

1812,2216 

2619 

3021 

j 108 

033424 

3826 

4227 

4628 

5029 

5430 

583016230 

6629 

7028 

; 1 09 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

• 207 

• 602 

• 998 

I 10 

041393 

1787 

2182 

257  6 

2969 

3362 

3755 

4148 

4540 

4932 

■111 

5323 

5714 

6105 

6495 

6885 

7275 

'7664 

,8053 

8442 

8830 

1 12 

9218 

9606 

9993 

• 380 

• 766 

1153 

,153811924 

2309 

2694 

1 13 

053078 

3463 

3846 

4230 

4613 

4996 

:5378  5760 

6142 

6524 

114 

6905 

7286 

7 666 

8046 

8426 

8805  9 1 85 j 9 563 

9942 

• 320 

■I  15 

060698 

1075 

1452 

1829 

2206 

2582:2958  3333 

3709 

4083 

,116 

4458 

4832 

5206 

5580 

5953 

6326:6699,7071  7443 

7815 

1 1 7 

8186 

8557 

8928 

9298 

9668 

■ • 58 

• 407 

• 776  1 145 

1514 

I 18 

071882 

2250 

2617 

2985 

3352 

37.1814085  4451,4816 

5182 

119 

5547 

5912 

6276 

6640 

7004 

7368/731,809418457 

8819 

r 1 20 

9181 

9543 

9904 

• 266 

• 626 

• 987;  1347. 170712067 

2426 

121 

0S2785 

3144 

3503 

3861 

4219 

4576,4934 

|529  115647 

6004 

122 

6360 

6716 

7071 

7426 

7781 

8136  8490  8845  9198 

9552 

123 

9905 

• 25S 

. 61  1 

• 963 

1315 

1667 

2018j2370i2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5 1 59:55  18 

5866162  15 

6562 

125 

6910 

7257 

7604 

7951 

8298 

S644  S990  9335|9681 

1026 

126 

100371 

07 1 5 

1059 

1403 

1747 

2091 

2434,2777 

3119 

3462 

127 

3804 

4 1 46 

4487 

4828 

5169 

55105851, 6 1 9 1,65  51 

6871 

128 

7210 

7549 

7888 

S227 

8565 

8903,9241  9579j99  16 

• 253 

129 

110590 

0926 

1263 

1599 

193.4 

2270  2605,2940,327 5 

3609 

1130 

3943 

4277 

4611 

4944 

5278 

561  1 

5943:627  6 

6608  6940 

1 131 

7271 

7603 

7934 

8265 

8595 

89  26925  6,9586:99  15 

0245 

13:2 

120574 

0903 

1231 

1560 

1S88 

2216 

25442871 

319S 

3525 

1 133 

3852 

4178 

4504 

4830 

5156 

5481 

ssoe^isi 

6456 

6781 

1 1 34 

7105 

7429 

7753 

8076 

8399 

8722|9045 

9368,9690 

12 

135 

130334 

0655 

0977. 

1298 

1619 

1939:2260 

2580^2900 

3219 

i 136 

3539 

3858 

41771449  6 

4814 

5133 

545  1 

5769,6086 

6403 

.137 

6721 

7037 

7354 

7671 

7987 

8303  8618 

8934:9249 

9564 

1 1 3S 

9S79 

• 194 

• 50S 

822 

1 136 

I 450, 

1763 

2076,2389 

2702 

139 

143015 

3327 

3630:3951 

4263 

457414885 

5196 

5507 

58181 

■ 1 40 

6128 

6438 

6748; 

7058 

7367 

7676j 

7985 

8294  S603 

891  lj 

! 141 

9219 

9527 

9835 

142 

• 449: 

* 756, 

1063 

1370, 

1676 

’982 

j 142 

152288 

2594 

2900r 

3205 

3510 

3815i4120 

4424,4728' 

5032 

i 1 43 

5336 

5640 

5943J6246 

6549 

6852 

7154 

7457, 

7759  8061 

; 1 44 

8362 

8664 

8965  9266 

9567 

986S! 

168 

* 469| 

769 

1068 

.145 

161368 

1 667 

1967 

2266 

2564 

2863!3161 

3461 

o 7 5 8s  1 

4055 

,146 

4353 

4650  4947| 

5244 

554 1 

5838  6134 

6430 

5726! 

147 

7317 

7613 

7908  8203 

8497 

8792 

90S6i93S0:9C7J. 

J96S 

i 1 48 

170262 

0555'0S48[ 

1141 

1434 

1724, 

2019 

2311, 

2 505 

2895 

U 49 

l 3186 

3478 

376914060 

4351 

464 1 ; 

4932J 

5-22155  12I5SO-: 

OF  NUMBERS, 


N 

0 

1 

2 

3 

4 

5 

1 6 

7 

1 8 

9 I 

15 

5 17609 

638 

667( 

3 695 

9 724 

8 7536 

(M 

00 

b- 

5 8113 

1840 

3689 

15 

8977 

926 

4 9555 

983 

3 12 

6 413 

j.  69: 

. 980:127: 

i 558 

15; 

181 844 

1212 

3 241, 

270t 

3 298 

5 3270 

355, 

3839 412: 

4407 

15, 

4691 

497 

5 5255 

554; 

3 582 

5 61081639 

6674:6956 

7239 

1 5^ 

7521 

780, 

3 8084 

8361 

5 864' 

7 8928l920! 

9490!9771 

. 5 1 

15! 

190332 

061: 

089S 

117 

145 

1730|20k 

2289 

2567 

2846 

156 

3125 

3401 

3681 

3955 

4237 

45  1414792 

5069 

5346 

5623 

157 

5899 

6171 

6453 

672S 

7006 

718117556 

7832 

8107 

8382 

15? 

8657 

8935 

9206 

9481 

9755 

. . 29 

303 

. 577 

850 

1124 

15S 

201397 

1 67C 

1943 

2216 

248S 

2761 

3033 

3305 

3577 

3848 

16C 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

161 

6826 

7096 

7365 

7634 

7804 

8173  8441 

8710 

8979 

9247 

162 

9515 

978? 

51 

• 319 

• 586 

853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

-2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579* 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8273 

8536 

8796 

9060 

9323 

9585 

9846 

166 

220108 

0370 

063! 

0892 

1153  1414 

1675 

1936 

2196 

2456 

16f 

2716 

2976 

3236 

3496 

3755|4015 

4274 

45334792 

5051 

168 

5309 

5568 

5826 

6084 

6342[6600 

6858 

71  15  7372 

7630 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682;9938 

. 193 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234(2488 

2742 

171 

2996 

3250 

3504 

3757 

401 1 

4264(4517 

4770  5023 

5276 

172 

5528 

5781 

6033 

6285 

6537 

6789  7041 

7292,7544 

7795 

173 

8046 

8297 

8548 

8799 

9049 

9299  9550 

9800, 

. 50 

. 300] 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

3293‘2541 

2790 

175 

3038 

3286 

3534 

3782 

4030 

427714525 

4772,5019 

5266 

176 

5513 

5759 

6006 

6252 

6499 

6745  6991 

7237 7482 

7728 

177 

7973 

8219 

8464 

8709 

8954 

9198 9443 

9687(9932 

. 176 

178 

250420 

0664 

0908 

1151 

1395 

163811881 

2 125*2368 

2610 

179 

2853 

3096 

3338 

358013822 

4064j4306 

4548(4790 

5031 

■180 

5273 

5514 

5755 

5996  6237 

6477  6718 

6958  7198 

7439 

181 

7679 

7918 

8158 

8398  8637 

8877  91  IS 

9355:9594 

9833 

182 

260071 

0310 

0548 

0787  1025 

1 263;  1 501 

1739  1976 

2214 

183 

2451 

2688 

2925 

3162 

3399 

3636  3873 

4109  4346 

4582 

184 

4818 

5054] 

if 90 

5525  5761 

599616232 

6467  6702 

6937 

185 

7172 

74% 

7641 

7875:81  10 

8344,8580 

8812  9046 

9279 

186 

9513 

974a 

0980 

213 

446 

679! 

912 

1 144  1377 

1609 

187 

271842 

2074 

W)6 

2538 

2770 

300 1 

233 

3464  3696 

3927 

188 

4158 

4389 

4620 

4850 

5081 

5311/542 

5772  6002 

6232 

189 

6462 

5692 

6921 

7151 

7380 

7609  7838 

3067  8296 

6525 

190 

8754 

3982 

9211 

3439 

9667 

9895  . 

123 

351  • 

578 

806 

191 

281033 

1261 

1488 

1715 

1942 

2 1 69|2396 

1622  2849 

3075 

192 

3301 

3527 

3753 

3979 

1205 

4431  4656 

1882  5107 

5332 

193 

5557 

5782 

6007 

5232 

6456 

6681 6905 

'130  7354 

7578 

194 

7802 

3026 

8249 

3473 

3696 

3920,9143  . 

>366  9589 

>812 

195 

290035 

)257 

0480 

3702 

3925 

1 147!  1 

369 

591,1813 

1034 

196 

2256 

2478 

2699 

3920 

3141 

3363  3584  . 

58044025 

1246 

197 

4466 

4687 

4907 

>127 

>347 

5567,5787 

>007  6226 

>446 

198 

6665 

5884 

’104 

’’323 

'542 

775X7979 

5198  8416 

3635 

1199 

885319071 

3289  5 

>507 

3725 

1943; 

161  - 

378  . 

595  . 

813' 

LOGARITHMS 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

203 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

899  1 1 9 204 

9417 

204 

9630 

9843 

. . 56 

268 

. 481 

. 693 

. 906 

1 1 1 8i  1330 

1542 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340J5551 

5760 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436  7 646 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522,9730 

9938 

209 

320146 

0354 

0562 

0769 

0977 

1 184 

1391 

1598  1805 

2012 

210 

2219 

2426 

2633 

283913046 

3252 

3458 

3665 

3871 

4077 

211 

4282 

4488 

4694 

4899  5105 

5310 

5516 

5721 

5926 

6131 

212 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

. . . 8 

.211 

214 

3304 1 4 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236l 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

216 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

. . 47 

246 

219 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1 830 

2028 

2225 

220 

2423 2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392  4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

222 

3353  6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

. . 54 

224 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

226 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7366 

7554 

7744 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9260 

9456 

9646 

229 

9835 

. . 25 

.215 

. 404 

. 593 

. 783 

. 972 

1161 

1350 

1539 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

231 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

232 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

234 

9216 

9401 

9587 

9772 

9958 

. 143 

•o»28 

.512 

. 698 

. S8S 

235 

371068 

1253 

1 437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

237 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

239 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

. . 30 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1656 

1837 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

242 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

243 

5606 

5785 

5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

244 

7390 

75  68 

7746 

7923 

8101 

8279 

8456 

8634 

8811 

8989; 

245 

9166 

934S 

9520 

9698 

9875 

. . 51 

. 228 

. 405 

. 582 

. 7591 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

247 

2697 

2873 

3048 

3224 

3400 

357 5 

3751- 

3926 

4101 

4277 ! 

248 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

602  5 i 

249 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

OF  NUMBERS 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

397940 

81  148287 

8461 

8634 

8808 

8981 

9134 

9328 

9501 

251 

9674 

9847 

. . 20 

. 192 

. 365 

. 538 

. 711 

. 883 

1056 

1228 

252 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

254 

4834 

5005  5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

255 

6540 

67106881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

256 

8240 

8410(8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

257 

9933 

. 102 

. 271 

. 440 

. 609 

. 777 

. 946 

1114 

1283 

1451 

258 

411620 

1788  1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

259 

3300 

3467 3635 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

260 

4973 

5140  5307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

261 

6641 

6807  6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

262 

8301 

8467  8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

263 

9956 

. 121 

. 286 

. 451 

. 616 

. 781 

. 975 

1110 

1275 

1439 

264 

421604 

1768 1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

265 

3246 

34103574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

266 

4882 

5045  5208 

5371 

5534 

5697 

5860 

6023 

6186 

6349 

267 

6511 

667416836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297  8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9752 

9914 

. . 75 

. 236 

. 398 

559 

. 720 

. 881 

1042 

1203 

270 

431364 

1525]  1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2909 

3130(3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

272 

4569 

4729  4888 

5048 

5207 

5367(5526 

5685 

5844 

6004 

273 

6163 

6322  6481 

6640 

6800 

6957|7116 

7275 

7433 

7592 

274 

7751 

7909 

8067 

8226 

8384 

8542 

8701 

8859 

9017 

9175 

275 

9333 

9491 

9648 

9806 

9964 

. 122 

. 279 

. 437 

. 594 

. 752 

276 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2209 

2166 

2323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

278 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

279 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8552 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

. . 95 

282 

450249 

0403(0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

283 

1786 

1940  2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

284 

33 1 8 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

285 

4845 

4997 

5150 

5302 

5454 

5606 

5758 

5910 

6062 

6214 

286 

6366 

6518  6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

288 

9392 

9543 

9694 

9845 

9995 

. 146 

. 296 

. 447 

. 597 

. 748 

289 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

292 

5383 

553.2 

5680 

5829 

5977 

6126 

6274 

6423 

65Yl 

6719 

293 

686SI7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

295 

9822 

9969 

. 116 

.263 

. 410 

. 55 7 

. 704 

. 851 

. 998 

1145 

296 

471292 

1488 

1535 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

297 

2756 

2903 

3049 

3195 

3341 

3487  3633 

3779 

3925 

4071 

298 

4216 

4362 

4508 

4653 

4799 

494415090 

5235 

53S1 

5526 

1299 

5671 

5816 

5962 

6107 

6252 

6397  6542 

6687 

6832 

6976 

LOGARITHMS 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

300 

477121 

7266 

741 1 

7555  7700 

7844 

7989 

8133 

8278 

8422 

301 

8566 

871 1 

8855 

8999  9143 

9287 

9481 

9575 

9719 

9863 

302 

480007 

0151 

0294 

043810582 

0725 

0869 

1012 

1156 

1299 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

^588 

2731 

304 

2874 

3016 

3159 

3302 

3445 

3587 

373i 

3872 

4015 

4157 

305 

4300 

4442 

4585 

4727 

4869 

501 ) 

5)53 

5295 

5437 

5579 

306 

5721 

5863 

6005 

6147 

6289 

6450 

6572 

6714 

6855 

6997 

307 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

84  lU 

308 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

98  i 8 

309 

9958 

99 

. 239 

. 380 

. 520 

661 

801 

941 

1081 

1222 

310 

491362 

1502 

1642 

1782 

1922 

2062 

~201 

2341 

248! 

2621 

311 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

312 

4155 

4294 

4433 

4572 

471) 

4850 

4989 

5128 

5267 

54.  '6 

313 

5544 

5683 

5822 

59  60 

6099 

6258 

6376 

6515 

6653 

6791 

314 

6930 

7068 

7206 

7344 

7485 

7621 

7759 

7897 

8035 

8173 

315 

8311 

8448 

8586 

8724 

886 

8999 

9137 

9275 

9412 

9550 

3)6 

9687 

9824 

9962 

. . 99 

236 

374 

511 

. 648 

785 

. 922 

317 

501059 

1196 

1 333 

1470 

1607 

744 

1880 

2017 

2154 

2291 

318 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

319 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

1878 

5014 

320 

5150 

5286 

5421 

5557 

5693 

5828 

5964 

6099 

6234 

6370 

321 

6505 

6640 

6776 

6911 

7046 

718) 

7316 

7451 

7586 

7721 

322 

7856 

7991 

8 126)8260 

8395 

8530 

8664 

8799 

8934 

9068 

323 

9203 

9337 

9471|9606 

9740 

9874 

. 9 

143 

277 

. 41  1 

324 

510545 

0679 

0813 

0947 

1081 

■215 

1349 

1482 

1616 

1750 

325 

1883 

2017 

215l|2284 

2418 

2551 

2684 

2818 

295  1 

3084 

326 

3218 

335  1 

3484 

361 7 

3750 

3883 

4016 

4149 

4282 

3414 

327 

4548 

4681 

4813  4946 

5079 

52!  1 

5344 

5476 

5609 

5741 

328 

5874 

6006 

6139,6271 

6403 

6535 

6668 

6800 

6932 

7064 

329 

7196 

7328 

746017592 

7724 

7855 

7987 

8119 

8251 

8382 

330 

8514 

8646 

8777,8909 

9040 

9 t 7 l 

9303 

9434 

9566 

9697 

331 

9828 

9959 

. . 90 

. 221 

• 353 

484 

615 

745 

. 876 

1007 

532 

521138 

1269 

1400!  1530 

1661 

1792 

1922 

2053 

2183 

2314 

333 

2444 

2575 

2705  2835 

2966 

3096 

3226 

3356 

3486 

5616 

334 

3746 

3876 

4006  4136 

4266 

4396 

4526 

4656 

4785 

4915 

335 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

336 

6339 

6469 

6598 

6727 

6856 

6985 

71  14 

7243 

7372 

7 50 1 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8531 

8660 

8788 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

72 

339 

530200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

135  I 

340 

1479 

1607 

1734 

1862 

1990 

2117 

2245 

2372 

2500 

-627 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5 1 67 

343 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

344 

6553 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

79 

204 

347 

540329 

0455 

0580 

0705 

0830 

0955 

10S0 

1205 

1 330 

1454 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

349 

2825 

2950 

307 4 

3199 

3323 

3447 

3571 

3696 

38201 

3944 

OF  NUMBERS. 


tN-l 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 v 

35>) 

544  '66 

4192  4316 

444o 

45  6 i 

4688 

4812 

4936 

5060 

5 183) 

351 

5307 

543 1 5555 

5678 

58)2 

592o 

5049 

5172 

6296 

6419 

362 

6543 

6666 

6789 

6913 

7036 

7 159 

7282 

7405 

7529 

7652 

353 

7775 

7898 

802 ! 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

354 

9003 

9 1 26 

9249 

9371 

*494 

9616 

9739 

9861 

9984 

196 

355 

550228 

035  1 

0473 

0595 

<717 

)840 

0962 

1084 

1206 

1328 

356 

145-1 

1572 

1694 

1816 

1938 

2060 

-2181 

2303 

2425 

2547 

357 

2668 

2790 

291  i 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

359 

5094 

5215 

5336 

5457 

5578 

5699 

7 820 

5940 

606! 

6182 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

361 

7507 

7627 

7748 

7868 

798§ 

8108 

8228 

8349 

8469 

8589 

362 

8709 

8829 

8948 

9068 

9188 

93')8 

9428 

9548 

9667 

9787 

363 

9907 

. 26 

146 

. 265 

385 

. 504 

. 624 

743 

. 863 

982 

364 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

i936 

2055 

2173 

365 

2293 

2412 

253  1 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

366 

3481 

3600 

3718 

3837 

3955 

4o74 

4192 

4311 

4429 

4548 

367 

4666 

4784 

4903 

5021 

5 1 39 

5257 

5376 

5494 

5612 

5730 

368 

5848 

5966 

6j84 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084' 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9257 

37  1 

9374 

949  1 

9608 

9725 

9882 

9959 

. . 76 

193 

. 309 

. 426 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

i476 

1592 

373 

1709 

1825 

1942 

2058 

2174 

229  1 

2407 

2523 

2639 

2755 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915' 

37  5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072' 

376 

5188 

5303 

5419 

5534 

5 650 

5765 

5880 

5996 

611.1 

6226 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

‘378 

7492 

7 6<>7 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525; 

,379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

95  55 

9669 

3 8>  > 

9784 

9898 

12 

. 126 

241 

. 355 

. 469 

. 583 

. 697 

.811) 

38  i 

58  >925 

U)39 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950; 

382 

2063 

2177 

229  1 

2404 

2518 

2631 

2745 

2858 

2972 

3085| 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218] 

384 

4331 

44  4 

4557 

4670 

4783 

4896 

5009 

5122 

523  5 

5348] 

385 

5461 

5 574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7 599) 

>387 

771  l 

7823 

7935 

S047 

8I60 

8272 

838>; 

849C 

8608 

8720 

388 

8832 

8944 

9056 

9167 

9279 

939  1 

9503 

9615 

9726 

9834 

389 

9950 

. 61 

173 

. 284 

. 396 

507 

. 619 

. 73C 

. 842 

. 953 

390 

591065 

1176 

1287 

1399 

1510 

162*1 

1732 

1843 

1955 

2066 

39! 

2177 

2288 

2399 

2510 

262  1 

2732 

2843 

2954 

3064 

3175 

392 

3286 

3397 

3508 

3618 

3729 

384C 

3950 

4061 

4171 

4282 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5 165 

5276 

5386 

39  4 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

■397 

8791 

8900 

900S 

91  19 

9228 

9337 

9446 

955  € 

9666 

9774 

398 

9883)9992 

101 

2 1 (. 

319 

428 

537 

646 

755 

864 

f-399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1951 

LOGARITHMS 


-N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400 

602060 

2169 

2277 

2286 

2494 

2603 

2711 

2819 

2928 

3036  • 

401 

3144 

325  3 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

•402 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

5089 

5197 

403 

530.5 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

69  19 

7026 

7133 

7241 

7348 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

407 

9594 

9701 

9808 

9914 

. - 21 

. 128 

. 234 

. 341 

447 

. 554 

408 

6 10660 

0767 

0873 

0979 

1086 

1 192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

267.8 

410 

2784 

2890 

2996 

3102 

3207 

33 1 3 

3419 

3525 

3630 

3736 

41 1 

3842 

3947 

4053 

4 1 59 

4264 

4370 

4475 

4581 

4686 

4792 

412 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

5^45 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6^90 

6895 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676|8780 

8884 

8989 

; 41  6 

9293 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

..32 

417 

620136 

0140 

0344 

0448 

0552 

0656 

0760 

0864 

0068 

1072 

418 

1 176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

5107 

5210 

422 

5312 

5415 

5518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

425 

8389 

8491 

8593 

8695 

8797 

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7845 

7890 

7934 

7979 

8024 

8068 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

.8782 

8826 

8871 

8916 

8960 

975 

9005 

9049 

9049 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

977 

9895 

9939 

9983 

. . 28 

. . 72 

. 117 

. 161 

. 206 

. 250 

294 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738  : 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

29511 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152  ' 

989 

5196 

5249 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

990 

5635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

3172 

3216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

3564  ;! 

3608 

3652 

997 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000  f9043 

1087 

998 

9131 

9174 

9218 

9261 

9305 

9348, 

9392 

9435  ’9479 

1522 

999 

9565 

9609 

9652 

9696 

9739 

9783. 

9826 

9870  19913 

1957 

Vol.  II  .3 


LOS.  SINES,  TANGENTS,  Nc. 


0 fieg. 


1 Leg. 


Sine,  i 

Cosine. 

Tang.  Cotang. 

Sine.  ) 

Cosine.  I 

Tang. 

Cot  an 1 

10-000000 

1 

8-241855 

9-999934 

8-241921 

1 1*758079*60 

6-463726  10000000 

6-463726  13-536274 

8-249033 

9-999932 

8-2491- (2 

11-750898 

59 

6-764756  1 

0-000000 

6-7647564  3-235  244 

8-256094 

9-999929| 

8-256 1 65 

11-743835 

58 

6-940847  1 

0-000000 

6-940847) 13-059153 

8-263042 

9-999927 

8-2631 15 

U -736885 

57 

7-065786  1 

0-000000 

7-0657861 12-934214 

8-269881 

9-999925] 

8-269956 

11-730044 

56 

7-162696 

0-000000 

7-162696  12-837304 

8-276114 

9-9999221 

8-276691 

11-723309 

55 

7-241877 

9-999999 

7-241878  12-758122 

8-283243 

9-999920 

8-283323  11-716677 

54 

7-308824 

9-999999 

7-308825  12-691 175 

8-289773 

9999918 

8-289856  11-710144 

53 

7-366816 

9-999999 

7-366817 

2 633183 

8-296207 

9-999915 

8-296292|ll-703708 

52 

7-417968 

9-999999 

7-417970 

12-582030 

8 302546 

9-999913 

8*3026-3411-697366 

51 

7-463726 

9-999998 

7-463727 

2-536273 

8-308794 

9-999910 

8-308884)  11  -69111 6 

5<i 

7-505118 

9-999998 

7-505120 

2-494880 

8-314954 

9-999907 

8-315046)  11 -684954 

49 

7-542906 

9-999997 

7-542909 

2-457091 

8-321027 

9-999905 

8-321 122jl  1-678878 

48 

7-577668 

9-999997 

7-577672 

12-422328 

8-327016 

9-999902 

8-3271 14)1 1-672886 

47 

7-609853 

9-999996 

7-609857 

12-390143 

8-332924 

9-999899 

8-333025)11-666975 

46 

7-639816 

9-999996 

7-639820 

12-360180 

8-338753 

9-999897 

8-338856]l  1-661 144 

45 

7-667845 

9-999995 

7-667849 

12-332151 

8-344504 

9-999894 

8-344610,11-655390 

44 

7-694173 

9-999995 

7-694179 

12-305821 

8-350181 

9*999891 

8-350289111-649711  !43 

7-718997 

9-999994 

7-719003 

12-280997 

8-355783 

9-999888 

8-355895  11-644105:42 

7-742478 

9-999993 

7-742484 

12-257516 

8-361315 

9-999885 

8-36145o|  1 1 • 638570)41 

7-764754 

9-999993 

7*764761 

1 ‘2-235239 

8-366777 

9 999882 

8-306895)  1 1 633105:40 

7-785943 

9-999992 

7-785951 

12-214049 

8-372171 

9-999879 

8-372292  11 -62770S,  39 

7-806146 

9-999991 

7-806155 

12-193845 

8-377499 

9-999876 

8-377622  11-622378,38 

7-825451 

9-999990 

7-825460) 12-1 74540 

8-382762 

9-999873 

8-382889 

11-617111,37 

7-843934 

9-999989 

7-843944  12-156056 

8-387962 

9-999870 

8-388092 

11*61 1908)36 

7-861662 

9-999989 

7-861674 

12-138326 

8-393101 

9-999867 

8-393234 

11-606766)35 

5 7-878695 

9-999988 

/ '870708 

12  121292 

8-398179 

9-999864 

8-398315 

11-601685  34 

- 7-S95085 

9-999987 

7-895099 

12-104901 

8-403199 

9-999861 

8-403338 

11-59666233 

8 7-910879 

9-999986 

7-910894 

12-089106 

8-408161 

9-999858 

8-408304 

U-591696.32 

l 7-926119 

9-999985 

7-926134 

12-073S66 

8-413068 

9-999854 

8-413213 

11-58678 

31 

0 7-940842 

9-999983 

7-940858 

12-059142 

8-417919 

9-999851 

8-418068 

11-581932,30 

1 7-955082 

9-999982 

7-955100 

12-044900 

8-422717 

9-999848 

8-422869 

11-577231  29 

2 7-968870 

9 999981 

7-968889 

12-031111 

8-427462 

9-999845 

8-427618 

ll -572382  28 

3 7-982233 

9-999980 

7-982253 

12-017747 

8-432156 

9-999841 

8-432315 

ll-567685;-27 

4 7-995198 

9-999979 

7-995219 

12-004881 

8-436800 

1 9-999838 

8-436962 

11-56303826 

5 8-007787 

9-999977 

8-007809 

11-992191 

8-441394 

9-999834 

8-441560 

11-558440  25 

6 8-020021 

9-999976 

8-020044 

11-979956 

8-445941 

9-999831 

8-446110 

11-553890  24 

7 8-C31919 

9-999975 

8-031945 

11  968055 

8-450440 

9-999827 

8-450613 

11-54938 

7 23 

8 8-043501 

9-999973 

8043527 

11-956473 

8-454893 

9-999824 

8-455070 

11-544930,22 

9 8-054781 

9-999972 

8-054809 

11-945191 

8-459301 

9-999820 

8-459481 

11-54051921 

0 8-065776 

9-999971 

8-065806 

11-934194 

8-463665 

9-999816 

8-463849 

11-536151  20 

1 8-O7650C 

9-999969 

8-076531 

11-923469 

8-467985 

9-999813 

8-46817 

2 

11-531 82S  19 

t2  8-086965 

9-999968 

8-086997 

1 1-913003 

8-47226.' 

9-999S0S 

8-472454 

1 1-527546: 13 

t3  8-09718C 

9-999966 

8-097217 

11-902783 

8-476495 

9-999805 

8-476693 

11 -523307!  17 

ti  8-107167 

9-999964 

8-107203 

11-892797 

8-48069; 

9-999801 

S-480892 

11 -51910S!  1 6 

t5  8-116926 

9-99996C 

8-116963 

11-883037 

8-484845 

9-999797 

8-485050 

1 1-514950;  15 

t6  8-126471 

9-999961 

8-12651C 

11-873490 

S-48896; 

9-99979-3 

8-48917 

-0 

11-51083014 

47  8-1358K 

9-999951 

8-135851 

11-864149 

8-49.-, 04( 

9-99979C 

8-493250 

11-50675013 

48  8-14495. 

9-999951 

8-14499E 

11-855004 

8-497075 

9-99978E 

8-497293 

11-502707:12 

49  8-15390" 

9-999956 

8-15395*/ 

11-846048 

8-501 08( 

9 999785 

8-501298 

11-49870211 

50  8-16268 

9-99995' 

8-162727 

11-837273 

8-50504. 

9-999771 

8 505267 

11-494733  10 

51  S 17128 

1 9 999951 

8171328 

11-828672 

8-50897- 

9-999774 

8-509200 

1 1-490S00  9 

52  8-1797 13|  9-999951 

) 8-17976. 

1 1-820257 

8-51286 

9-999761. 

8-513098 

11-486902  S 

53  8-187985|  9-99994 

8-188031 

11-811964 

8-51672 

9-99976; 

8-516961 

11-483039  7 

54  8-1961021  ,9-99994 

5 8-196151 

11-803844 

8-52055 

9-99976 

8-520790 

11 -479210,  6 

55  8-204070,  9-99994 

4 8-2(1412 

11-765874 

8-52434 

3 9-99975/ 

85245S6 

11-475414!  5 

56  8-21 1895 1 9-99994 

2 8-21125 

11-788047 

8-52810 

2 9-99975. 

8-52S349 

11-471651  4 

57  8-2195811  9-99994 

0 8-21964 

1 11-780359 

8-53182 

3 9-999741 

8-55208011-467920'  3 

58  8-227134  9-99993 

8 8-22719 

5 1 1-772805 

8 53552, 

3 9-99974- 

8*5357 

'9  11-464221  2 

59  8-234557  9-99993 

6 8-23462 

1 11  765379 

8-539186  9-999741 

8-539447 

ill -460553  1 

60  8-241855  9-99993 

4 8-24(92 

1 11-758079 

8-542S19  9-99973 

8-5430S4|l  1-45691 6 0 

Cosine 

Sine. 

Cotan. 

| Tang. 

Cosine. 

Sine. 

Cotan 

| T sug. 

SS  Deg. 


LOG.  SINES,  TANGENTS,  &c. 


2 JDe 

S- 

3 Deg. 

i Sine. 

Cosi  .,i. 

Tan£. 

Cotang. 

Sine. 

| Cosine. 

I Tang 

| Cotang. 

j 

I L 

1 - -542819 

9-999735 

8*5430;4 

,11-456916 

8-71 8SU0 / 9-9994i  - 

8-719396  11-28060- 

'160 

i 

8-546422 

9-999731 

8-54669 

11-453309 

8-721204 

9-99939! 

8-72180 

',(11-27819. 

59 

2 

8-549995 

9-999726 

8-55026 

1 1-449732 

8-72359£ 

9-999391 

8-7242(1- 

U-27579f 

58 

3 

8553539 

9-999722 

8-55381 

•1-446183 

8-725975 

9-999384  8-72658! 

11-27341! 

57 

4 

8-557054 

9-999717 

8-55731 

i ’442664 

8-728337 

9-99937? 

8-72895! 

11-271041 

56 

5 

8-560540 

9-99971. 

8-56082 

11439179 

8-73068? 

9-999371 

8-73131 

11-26868! 

55 

6 

8-563999 

9-999708 

8-56429 

1 1^35709 

8-733027 

9 999364 

8-73366! 

11-26633/ 

54 

7 

8-567431 

9-999704 

8-567727 

11-432273 

8-73535-j 

9-999357 

8-735996 

11-26400- 

53 

8 

8 5r0836 

9-999699 

8-57113/ 

1 1 -42*863 

8-737667 

9-99935C 

8-738317 

11  2616S. 

52 

9 

S-574214 

9-999694 

8-57452C 

11-425180 

8-739969 

9-999343 

8-74062! 

ll-25937-i 

51 

10 

8-577506 

9-999689 

8-57787/ 

11-422123 

8-742259 

9-999336 

8-742925 

11-25707S 

50 

it 

8-580892 

9-999685 

§-581208 

11-418792 

8-744536 

9-999329 

8-745207 

11.254795 

49 

12 

8-584193 

9-999680 

8-584514 

11-415486 

8-746802 

9-999322 

8-747479 

11-252521 

48 

IS 

8-5S7469 

9*999675 

8-587795 

11  412205 

8-749055 

9-999315 

8-74974  C 

11-25026C 

47 

14 

8 590721 

9-999670 

8-591051 

11-408949 

8-751297 

9-999308 

S-75198911 1 -24801 1 

46 

15 

8-593948 

9-999665 

8-594283 

11-405717 

8-753528 

9-999301 

8-754227  i 11  -245773 

45 

16 

8-597152 

9-999660 

8-597492 

11-409508 

8 755747 

9-999294 

8-756453  11-243547 

44] 

1? 

8-600332 

9-999655 

8-600677 

11-359323 

8-757955 

9-999287 

8-758668  11-241332 

43 

18 

8-603489 

9-999650 

8-603839 

11396161 

8-760151 

9-999279 

8-760872,1 1 2391 28 

42 

19 

8.606623 

9-999645 

8-606978 

11-393022 

8-762337 

9-999272 

8-763065 'll -236935 

4l  | 

20 

8-609734 

9-999640 

8-610094 

'll -389909 

8-764511 

9-999265 

8-765246  1 1 -234754 

40a 

21 

8-612823 

9-999635 

8-61318? 

II -3868 11 

S-766675 

9-999257 

8-76741 7 'H-232583 

39 1 

22 

8-615891 

9-999529 

8-616262 

11-383738 

8-768828 

9-999250 

8-769578  11 -2304  2‘. 

38 1 

23 

8-618937 

9-999624 

8-619313 

1 1*380687 

8-770970 

9-999242 

8-771727 

11  -228273 

370 

24 

8-621962 

9-999619 

8-62i343 

11  377657 

8-773101 

9-999235 

8-773866111-226134 

36  H 

25 

8-624965 

9-999614 

3625352 

11-374648 

8-775223 

9-999227 

8-775995 

1 1 -224005 

35  a 

26 

8 627948 

9-999608 

8628340 

11-371660 

8-777333 

9-999220 

8-778114 

11-221886 

348 

27 

8-630911 

9-99960? 

8-631308 

11-368692 

8-779434 

9-999212 

8-780222 

11-219778 

33 1 

28 

8-633854 

9-99959 

8-634256 

11-365744 

8-781524 

9-999205 

8-782320 

11-217680 

321 

29 

8-636776 

9-999572 

8-63/584 

11-362816 

8-783605 

9-999197 

8-784408 

11  -215592(31 8 

30 

S 639680 

9-995>86 

8-640093 

11-359907 

8-785675 

9 999189 

8-786486 

ll-213514j30S 

31 

8-642563 

9-9^581 

8 642982 

11-357018 

8-787736 

9-999181 

8-788554 

11-211446 

298 

32 

8-645428 

9-99575 

8-645853 

11-354147 

8 789787 

9-999174 

8-790613 

11-209337 

25  ■ 

33 

8-648274 

<999570 

8-648704 

11-351296 

8-791828 

9-999166 

8-792662 

11-207338 

27  B 

54 

8-651102 

J-999564 

8-651537 

11-348463 

8793859 

9-999158 

8-794701 

11-2052S9 

26  j 

35 

8-653911 

9-999558 

8-654352 

11-345648 

8-795881 

9-999150 

8-796731 

11-205269 

25 

36 

8-65670 

9-999553 

8-657149 

11-342851 

8-797894 

9-999142 

8-798752 

H-C01248 

241 

1 37 

8-659  /5 

9-999547 

8-659928 

11-340072 

8-799897 

9 999134 

8-800763 

11-199237 

23 

,38 

8-6£'230 

9-999541 

8-662689 

11-337311 

8-801892 

9-999126 

8-802/foi 

11-197235 

22 

'39 

8-6,4968 

9-999535 

8-665433 

11-334567 

8-803876 

9-999118 

8 804758 

11-195242 

21 

40 

8,67689 

9-999529 

8-668160 

11-331840 

8-805852 

9-999110 

fl  806742 

11  193258 

20 ; 

41 

8670393 

9-999524 

8-670870 

11-329130 

8-807819 

9-999102 

8-808717 

11-191283 

19 

42 

/■673080 

9-999518 

8-673563 

11-326437 

8-809777 

9-999094 

8-810683 

11-189317 

IS 

! 43 

8-675751 

9 999512 

8 676239 

11-323761 

8-811726 

9-999086 

8-812641 

11-187359 

,7 1 

44  8-678405 

9-999506 

8-678900 

11-321100 

8-813667 

9-999077 

8-814589 

U-185411 

16 

4 

8-681043 

9-999500 

8-681544 

11-318456 

8-815599 

9 999069 

8-816529 

11-183471 

15. 

4) 

8-683665 

9-999493 

8-684172 

11-315828 

8-817422 

9-999061 

8-818461 

11-181539 

:4i 

*7 

8-686272 

9-999487 

8-686784 

11-313216 

8-819436 

9-999053 

8-820384 

11-1796'6 

3 

iS 

8-688863 

9-999481 

8-689391 

11-310619 

8-821343 

9-999044 

8-822298 

li  -177/02 

2< 

49 

8-691438 

9-999475 

8-691963 

11-308037 

8-823240 

9-999036 

8-824205 

1 1 175795 

1 

50 

8-693998 

9-999469 

8-694529 

11-305471 

8 825130 

9-999027 

8-893103 

1-173897 

0 

51 

8-696543 

9-999463 

8 697081 

11-302919 

8-827011 

9-999019 

8-527992 

1-172008 

9 

52 

8-699073 

9-999456 

8-699617 

11-300383 

8-828884 

9-9990/0 

S-829S74 

1-170126 

8 

55 

8-701589 

9-999450 

8-702139 

11-297861 

8-830749 

9-999002 

8-831748 

1-168252 

7 

5- A 

8-704090 

9-999443 

8-704646 

11-295354 

8-832007 

9-998993 

8-833613 

1-166387 

6j 

55 

8-70657 7 

9-999437 

8-707140 

11-292860 

8-834456 

9-998984 

8-835471 

1 -1 64529 

5 

56 

5-709049 

9-999431 

8-709618 

11-290382 

8 836297 

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9 326700 

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9-327281 

9-989910 

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9-360752 

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9 372499  10-627501 

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9-327862 

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9-337919 

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9-988223  9-373064!  10626936 

4.3 

18 

9-328442 

9-989915 

9-338527 

10-661473 

9-361822 

9 988193 

9-37362940-626371 

42 

19 

9-329021 

9-989817 

9-339133 

10-660867, 

9-362356 

9-988163 

9-374193  10-625807 

41 

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9-329599 

9-989830 

9-339739 

10-6602611 

9-362889 

9-988133 

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40 

21 

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9-989832 

9-340344 

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9-363422 

9 988103 

9-375319 

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39 

22 

9-330753 

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9-340948 

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9-331329 

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9-341 552 

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9-331903 

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9 989751 

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9-533624 

9-989665 

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9-987922 

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33 

28 

9-334195 

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32 

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9-33476 7 

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9-345157 

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31 

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9:372373 

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9-  87217 

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9-259313 

10-640687 

9-380113 

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9 3929S9 

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9-987092 

9 393531 

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6 

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9-988869 

9-360474 

10-639526 

9-3S1134 

9-9S7061 

9-39+073; 

10-605927 

5 

56 

9-349893 

9-988840 

9-361053 

10-63S947 

9-381643 

9-9S7030 

9-394614  10-6053S6 

4 

57 

9-350443 

9-988811 

9-361632 

10-638368 

9-382152 

9-9S6998 

9-395154  10-604846; 

58 

9-350992 

9-988782 

9-362210 

10-637790 

9382661 

9-986967 

9-39569  V. 

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- 

59 

9-351540 

9-988753 

9-362787 

10-637213 

9-383168 

9-98''936 

9-396253 

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60 

9-352088 

9-988724 

9-363364 

10-536636 

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Cosine. 

Sine 

Cotan. 

Tang. 

Cosine  1 

Sine. 

' 0 an. 

Tang.  1 

77  Deg. 

76  Den. 

LOG.  SINES,  TANGENTS,  &<s. 


if 

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—7? 

Deg. 

- 

| Sine. 

| Cosine. 

Tang. 

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Cosine. 

I Tang. 

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> 9-38367J 

9-986904 

9-396771 

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9-428052 

10-571941 

60 

9-3841 85 

9-9S6S7„ 

9-397309  10-602691 

9-413467  9-9849 1C 

9-428558 

10-5  71 44-. 

59 

9-38468? 

9-986841 

. 9-397846  10-602154 

9-413938 

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9-429062 

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3 

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9-986809 

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9-41440S 

9-984845 

9-429566 

10-570434 

57 

4 

9-385697 

9 986778 

9-398919 

10-601081 

9-41 487S 

9-984808 

9-430071 

10-56993C 

56 

5 

9-386201 

9-986746 

9-399455 

10-600545 

9-415347 

9-98477-1 

9-430576 

10  569427 

55 

6 

9-386704 

9-986714 

9-599990 

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9 415815 

9-98474C 

9-431075 

10-568925 

54 

7 

9-387207 

9-986683 

9-400524 

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9-416283 

9-984706 

9*431577 

10-568423 

53 

9-387709 

9-986851 

9-401058 

10-598942 

9 410751 

9-984672 

9 432079 

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52 

9 

9-388210 

9-986619 

9-401591 

10-598409 

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9-432580 

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51 

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9-984603 

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50 

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9 986555 

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48 

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9-419079 

9 984500 

9-434579 

10-565421 

47 

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9-986459 

9-404249 

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9-984466 

9 43:. 078 

10-564922 

46 

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9-391206 

9-986427 

9-404778 

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9-420007 

9-984432 

9-435576 

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45 

15 

9-391703 

9-986395 

9-405308 

10-594692 

9-420470 

9-984397 

9-436073 

10-563927 

44 

17 

9-392199 

9-986363 

9-405836 

10-594164 

9-420933 

9-984363 

9-436570 

10-563430143 

18 

9-392695 

9-  -86331 

9-406364 

10-593636 

9-421395 

9-9a4328 

9-437067 

10-562933i42 

19 

9-393191 

9-986299 

9-406892 

10-593108 

9-421857 

9-984294 

9-437563 

10-562437I41 

20 

9-393685 

9-986266 

9-407419 

10-592581 

9-422318 

9-984259 

9-438059 

10-561941  40 

21 

9-394179 

9-986234 

9-407945 

10  592055 

9-422778 

9-984224 

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10-561446139 

22 

9-394673 

9-986202 

9-408471 

10-591529 

9-423238 

9-984190 

9-439048 

10-560952 

38 

23 

9-395166 

9-986169 

9-408996 

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9-984155 

9-439563 

10-560457 

37 

24 

9-395658 

9-986137 

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9-440036 

10-559964 

36 

25 

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9-980104 

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35 

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9-425073 

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32 

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9-398111 

9-985974 

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9-983946 

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10-557503 

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9-398600 

9 985942 

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30 

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9-413179  1 0-586821 1 

9-427354 

9-983875 

9-443479 

10-556.521 

29 

32 

9-399575 

9*985876 

9-413699  10-586301 

9-427809 

9-9S3840 

9-443968 

10-556032 

28 

33 

9-400062 

9-985843 

9-414219  10-5857811 

9-428263 

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27 

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9 428717 

9-983770 

9-444947 

10-555053 

26 

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9-415257  10-584743 

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9-983735 

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25 

36 

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9-985745 

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9-429623 

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9-445923 

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9-450075 

9 983664 

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9-402489 

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10-583190 

9-430527 

9-983629 

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10-553102 

22 

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9-402972 

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9-417326 

10  582674 

9-430978 

9-983594 

9-447384 

10-552516 

21 

10 

9-403455 

9-985613 

9-417842 

10-582158 

9-431429 

9*983558 

9 447870 

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10-581642 

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19 

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9-404420 

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9-985514 

9-419387 

10  580613 

9-432778 

9-983452 

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10*550674 

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9-985381 

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9-983309 

9-451260 

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13 

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9-985347 

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9 451743 

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10  547775 

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9-985280 

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9-983166 

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9-985213 

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10-576007 

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9-983130 

9-453668 

10-546332 

53 

9-409682 

9 985180 

9-424503 

10-575497 

9-437242 

9-983094 

9-454148 

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54 

9-410157 

9-985146 

9-425011 

10-574989 

9-437686 

9-983058 

9-454628 

10*545372 

6 

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9-410632 

9-985113 

9-425519 

i 0 574481 

9-438129 

9-983022 

9-455107 

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9-411106 

9-985079 

9-426027 

10-573.-73 

9-438572! 

9-982986 

9-455586 

10-544414 

57 

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9-985045 

9*426534 

10  5734691 

9-439014; 

9-982950 

9-456064 

10-543936 

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58 

9-412052 

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9-427041 

10-572959j 

9-433456 

9-982914 

9-456542 

10-543458 

2 

59 

9-412524 

9 984978 

9-427547 

10-572453 

9-439397. 

9-982878 

9-457019 

10-542981 

1 

60 

9-412996 

9-984944 

9-428052 

10-571948. 

9-440338, 

9-982842' 

9-457496 

10-542504 

0 

Cosine. 

Sine. 

Cotan. 

Tang,  j 

Cosine,  j 

Sine. 

Cotan. 

Tang. 

~ 

75  D.-g. 

74  Deg. 

a 


l»g„  Sines,  tangents,  &c. 


“T^TTel  "iTff" 


j | Sine.  ( Cosine. 

Tans-  j Cotang.  ! 

Sine. 

Josine.  | 

Tang.  | Cota; 

0 9 440338 

1-982842 

9-457496110-542504 

9*465935 

9-980596 

9-4S5339!  1 u-514661  • 

1 9440778 

3-982805 

>45797.1  10-542027 

9-466348 

9-980558 

9-485791  1 

0-514209  .'9 

2 9441-218 

3-982769 

9-458449  1 

0-541551 

9-466761 

9-980519 

9-486242  1 

i 513758  53 

3 9 441658 

3-982733 

9-458925  l 

0-541075 

9-467173 

9-980480 

9-486693  1 

.'*513307  5 7 

4 9-442096 

9-982696 

9-459400  l 

0-540600; 

9-467585 

9-980442 

9-487143  l 

0-512857  5 

5 9-442535 

9-982660 

9-459875  1 

0-540125 

9-467996 

9-98"403 

9-487593  10-5124)7  5 

5 

6 9-442973 

9-982624 

9-460349  l 

0-539651 

9-468407 

9-980364 

9-488043  10-511957  54 

7 9-443410 

9-982587 

9-460823  1 

0-539177 

9-468817 

9-980325 

9-488492 

0-511508  5 

8 9 443847 

9-982551 

9-461297 

0-538703 

9-469227 

9-980286 

9-488941 

0-511059  5 

9 9-444284 

9-982514 

9-461770 

0-538230 

9"469637 

9-980247 

9-489390 

0-5 10610: 51 

10  9-444720 

9-982477 

9-462242 

0-537758 

9-470046 

9-980208 

9-489838 

0-510162,50 

11  9-445155 

9-982441 

9-462715 

0-537285 

9-470455 

9-980169 

9-490286 

L0  50971 4'49 

12  9-445590 

9-982404 

9-463186 

0-536814 

9-470863 

9-980130 

9-490733 

0-509267|48 

13  9-446025 

9-982367 

9-463658 

0-536342 

9-471271 

9-980091 

9 491180 

10-508820  47 

14  9-446459 

9-982331 

9-46412S 

0-535872 

9-471679 

9-9S0052 

9-491627 

0-50837346 

15  9-446893 

9-982294 

9-464599 

10-535401 

9-472086 

9-980012 

9-492073 

10-507927  45 

1C  9-447326 

9-982257 

9-465069 

10-534931 

9-472492 

9-979973 

9-492519 

10-507481144 

17  9-447759 

9-982220 

9-465539 

10-534461 

9-472898 

9-979934 

9-492965 

10.507035  43 

18  9-448191 

9-982183 

9-466008 

10-533992 

9-473304 

9-979895 

9-493410 

10.506590  42 

19  9-448623 

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9-466477 

10-533523 

9-473710 

9-979855 

9-493854 

10-506146, 

ii 

20  9-449054 

9-982109 

9-466945 

10-533055 

9-474115 

9-979816 

9-494299 

10-505701  [4o 

§21  9-449485 

9-982072 

9-467413 

10-532587 

9-474519 

9 979776 

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L0-  505257 

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§22  9-449915 

9-982035 

9-467880 

10-532120 

9-474923 

9-979737 

9-495186 

10-504814:38 

§23  9-450345 

9-981998 

9-468347 

10-531653 

9-475327 

9-979697 

9-495630 

10  504370 

57 

§24  9-450775 

9-981961 

9-468814 

10-531186 

9-475730 

9-97i)658 

9-496073 

10-503927 

36 

§25  9-451204 

9-981924 

9-4692SO 

10-530720 

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54 

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9-98 '849 

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28  9-452488 

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31  9-453768 

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9-979380 

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29 

32  9-454194 

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33  9 454619 

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40  9-457584 

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9-48842 

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10-489946 

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9-48881- 

1 9-97832! 

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5 9-48443 

5 10-515565 

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9-48959. 

9-978247 

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60  9 46593 

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9-97820f 

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0 

| Cosine. 

Sine. 

Cotan. 

Tang. 

1 Cosine. 

Sine. 

Cotan. 

J Tang. 

73 

I il'O- 

72  Deg. 

LOG.  SINES,  TANGENTS,  &c< 


7 ^ 


' 

| Sine. 

Cosine. 

j Tang. 

Cotang. 

Sine. 

1 Cosine. 

Tang. 

Cotang. 

€ 

9-489982 

9-978206 

9-511776 

10-488224 

9-512642 

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9-536972 

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9-490371 

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9-512206 

10-487794 

9-513009 

9-975027 

9-5373S2 

10-462618  59 

2 

9-490759 

9-978124 

9-512635 

10-487365 

9-513375 

9-975585 

9-537792 

10-462208158 
10-461798  57 

3 

9-491147 

9-978083 

9-513064 

10  486936 

9-513741 

9-975539 

9-538202 

4 

9-491535 

9-978042 

9-513493 

10-486507 

9-514107 

9*975496 

9-538611 

10-461389156 

5 

9-491922 

9-978001 

9-513921 

10-486079 

9-514472 

9-975452 

9-539020 

10-4609  80155 

6 

9-492308 

9-977959 

9-514349 

10-485651 

9-514837 

9-975408 

9-539429 

10-46057l|54 

7 

9-492695 

9-977918 

9-514777 

10-485223 

9-515202 

9-975365 

9-539837 

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8 

9-493081 

9-977877 

9-515204 

10-4S4796 

9-515566 

9-975321 

9-540245 

10-459755 

52 

9 

9-493466 

9-977835 

9-515631 

10-484369 

9*515930 

9-9752 77 

9-540653 

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51 

10 

9-493851 

9-977794 

9-516057 

10-483943 

9-516294 

9-975233 

9 541061 

10-458939 

50 

11 

9-494236 

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9-516657 

9-975189 

9-541468 

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49 

12 

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9-516910 

10-483090 

9-517020 

9-975140 

9-541875 

10-458125 

48 

13 

9-495005 

9-977669 

9-517335 

10-482665 

9-517382 

9-975101 

9-542281 

10-457719 

47 

14 

9-495388 

9 977628 

9-517761 

10-482239 

9-517745 

9-975057 

9-542688 

10-457312 

46 

15 

9-495772 

9-977586 

9-518186 

10-481814 

9-518107 

9-975013 

9-543094 

10-456906 

45 

16 

9-496154 

9-977544 

9-518610 

10-481390 

9-518468 

9-974969 

9 543499 

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44 

17 

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9-977503 

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9-518829 

9-974925 

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43 

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9-496919 

9-977461 

9-519458 

10-480542 

9-519190 

9-974880 

9-544310 

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42 

19 

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9-519551 

9-974836 

9-544715 

10-455285 

41 

20 

9-497682 

9-977377 

9-520305 

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9-519911 

9-974792 

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40 

21 

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9-977335 

9-520728 

10479272 

9-520271 

9-974748 

9-545524 

10-454476 

39 

22 

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9-977293 

9-521151 

10-478849 

9-520631 

9-974703 

9-545928 

10-454072 

38 

23 

9-498825 

9-977251 

9-521573 

10-478427 

9-520990 

9-974659 

9-546331 

10-453669 

37 

24 

9-499204 

9-977209 

9-521995 

10-478005 

9-521349 

9-974614 

9-546735 

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36 

25 

9-499584 

9-977167 

9-522417 

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9-521707 

9-974570 

9-547138 

10-452862 

55 

26 

9-499963 

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9-522838 

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9-522066 

9-974525 

9-547540 

10-452460;34 

‘27 

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9-977083 

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9-974481 

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10  450057133 

28 

9-500721 

9-977041 

9-523680 

10-476320 

9-522781 

9-974436 

9-548345 

10-451655 

32 

29 

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9-524100 

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9-974391 

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31 

30 

9-501476 

9 976957 

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10-475480 

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9-974347 

9-549149 

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30 

31 

9-501854 

9-976914 

9-524940 

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9-523852 

9-974302 

9-549550 

10-450450 

29 

32 

9-502231 

9-976872 

9 525360 

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9-524208 

9-974257 

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10-450049 

28 

33 

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9-524-564 

9-974212 

9-550352 

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27 

34 

9 502984 

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10-473803 

9-524920 

9-974167 

9-550752 

10-449248 

26 

35 

9-503360 

9-976745 

9-526615 

10-473385 

9-525275 

9-974122 

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25 

36 

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9-974077 

9 551552 

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24 

37 

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22 

21 

39 

9-504860 

9-976574 

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9-973942 

9-552750 

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9-505234 

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9-528702 

10-471298 

9-527046 

9-973897 

9-553149 

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20 

41 

9-505608 

9-976489 

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10-470881 

9-527400 

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9*553548 

10  446452 

19 

42 

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9-976446 

9-529535 

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9-973807 

9-553946 

10-446054 

18 

43 

9-506354 

9-976404 

9-529951 

10-470049 

9-528105 

9-973761 

9-554344 

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17 

44 

9-50672 7 

9-976361 

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9-528458 

9-973716 

9-554741 

10-445259 

16 

45 

9-507099 

9-976318 

9-530781 

10-469219 

9-528810 

9-973671 

9-555139 

10-444861 

1 5 

46 

9-507471 

9-976275 

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9-529161 

9-973625 

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1 0*444464 

14 

47 

9-507843 

9-976232 

9-531611 

10-468389 

9-529513 

9-973580 

9-555933 

10-444067 

1.3 

48 

9-508214- 

9-976189 

9*532025 

10-467975 

9-529864 

9-973535 

9-556329 

10-443671 

12 

49 

9-508585 

9-976140 

9-532439 

10-467561 

9-5S0215 

9-973489 

9-556725 

10-443275 

11 

50 

9-508956 

9-976103 

9:532853 

10-467147 

9-530565 

9-973444 

9-557121 

10-442879 

10 

51 

9-509326 

9-976060 

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10-466734 

9-530915 

9-973398 

9-557517 

1 0-442483 

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52 

9-509696 

9-976017 

9-533679 

10-466321 

9-531265 

9-973352 

9*557913 

10-442087 

s 

55 

9-510065 

9-975974 

9-534092 

10-465908 

9-531614 

9-973307 

9-558308 

10-441692 

7 

54 

9-510434 

9-975930 

9-534504 

10-465496 

9-531963 

9-973261 

9*558703 

10-441297 

6 

55 

9-510803 

9-9758S7 

9-534916 

10-465084 

9-532312 

9-973215 

9-559097 

10-440903 

5 

56 

9 511172 

9-975844 

9-535328 

10-464672 

9-532661 

9-973169 

9-559491 

10-440509 

4 

57 

9-511540 

9-975800 

9-535739 

10-464261 

9-533009 

9-973124 

9-559885 

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3 

58 

9-511907 

9-975757 

9-536150 

10-463850 

9 533357 

9 973078 

9-560279 

10-439721 

o 

59 

9-512275 

9-975714 

9-536561 

10-463439 

9-533704 

9-973032 

9-560673 

10-439327 

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60 

9-512642 

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9-536972 

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9-5nlu66 

10-438934 

0 

Cosine.  1 

Sine.  1 

Cotan. 

Tang. 

Cosine. 

Sine.  | 

Cotan. 

Tang. 

/ 

71 

Deg. 

70  Deg 

m 

LOG.  SINES  TANGENTS,  &c. 


1“ 

2i 

— 

21  Dt,.. 

S'lie  | Cosine. 

Tang-  1 

Coiang. 

Sine 

Cosine.  1 

Tang.  iColang.  1 

§ 0 

9-5340521 ' 

9-972986 

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0*438934 

9*554329 

9-970152 

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9-534399 

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( -415445  59 

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9.972894 

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0-438149 

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9-970055 

9-584932 

0-415068!  l,8 

9-535092, 

9-972848 

9-562244 

0-437756 

()  555315 

9-970006 

9-5853u9 

0-414691  57 

4 

9*535438i 

9-972802 

9-562630 

0-437364 

9-555643 

9-969957 

9-585686 

0-414314  56 

5 

9-535783 

9-972755 

9-563028 

0-436972 

9-555971 

9-969909 

9-586062 

0-413938  55 

6 

9-536l29j 

9-972709 

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0-436581 

9-556299 

9-969860 

9-586439 

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7 

9-536474’ 

9-972663 

9-563811 

10-436189 

9-556626 

9-969811 

9-586815 

0-413185  53 

8 

9-536818 

9-972617 

9 564202 

10-435798 

9-556953 

9-969762 

9-587190 

10-412810  52 

9 

9-537163 

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9-587566 

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10 

9-537507 

9-972524 

9-564983 

10*435017 

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9-969665 

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10-412059150 

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9-537851 

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9-969616 

9-588316 

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9-538194, 

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9-969567 

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9-972385 

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10-433847 

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9 969518 

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9-538880! 

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9-566542 

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9-539223 

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9-566932 

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9*559234 

9-969420 

9-589814 

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16 

9-539565 

9-972245 

9-567320 

10-432680 

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9-969370 

9-590188 

10-4098124i 

17 

9-539907 

9-972198 

9-567709 

10  432291 

9-559883 

9-969321 

9*59056*2 

10-4O943SI43 

18 

9-540249 

9-972151 

9 568098 

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9-590935 

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19 

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10-431514 

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9-591508 

10-408692' 41 

20 

9-540931 

9-972058 

9-568873 

10-431 127 

9*560855 

9-969173 

9-591681  10-408319  40 

21 

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9-972011 

9-569261 

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9-561178 

9 969124 

9-592054  10-407946  39 

22 

9-541613 

9-971964 

9-569648 

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9-969075 

9*59  j426  (0*407574  38 

9-541953 

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24 

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9 562146 

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25 

9-542632 

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26 

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9-971776 

9-571195 

10-428805 

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9-968877 

9-5939 14|  10-406086  34 

27 

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29 

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9-545000 

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9-565356 

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1 403122  26 

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9-54601 1 

9-971351 

9-574660 

10-425340 

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9-968429 

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8 36 

9-546347 

9-971303 

9-575>-44 

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9-554329 

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Sine 

Cntsn. 

Tang. 

Cosine. 

Sine. 

1 Cotan. 

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69  Deg. 

68  Deg. 

LOG.  SINES,  TANGENTS,  kc. 


22  Deg  23  Deg. 


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Sine. 

Cosine. 

Tang. 

Cotang. 

Sine.  1 

Cosine.  | 

Tang. 

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0 

9-573575 

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9-606410 

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9-627852 

10-372148 

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9-573888 

9 967115 

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59 

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9 574200 

9-967064, 

9-607137 

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9-59*2473 

9-963919 

9-628554 

10  .371446 

58 

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9-574512 

9-967013 

9-607500 

0-392500 

9-592770 

9-963865 

9-628905 

10-371095 

4 

9-574824 

9-966961 

9-607863 

10  392137 

9-593067 

9-963811 

9-629255 

10-370745 

56  i 

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9-575136 

9-966910 

9-608225 

10-391775 

9-593363 

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9-629606 

10-370394 

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9-608588 

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9 593659 

9-963704 

9-629956 

10-370044 

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9-575758 

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9-608950 

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9-630306 

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9 5948+2 

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10-368645 

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9-632750 

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9.596609 

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10-365510 

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9-579777 

9-966136 

9-613641 

10-386359 

9-597783 

9-962945 

9-634838 

10-365102 

40 

21 

9-580085 

9-966085 

9-614000 

10-386000 

9-598075 

9-962890 

9-6351S5 

10-364815 

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9-580392 

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9-614359 

10-385641 

9-598368 

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9-035532 

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9-598660 

9-962781 

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9-965929 

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9 965824 

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9-599536 

9-962617 

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27 

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9-599827 

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35 

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9-582229 

9 965720 

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10-383491 

9-600118 

9-962508 

9-637611 

10-362389 

32 

29 

9-582535 

9-96566S 

9-616867 

10-383133 

9-6004U9 

9-962453 

9-637956 

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31 

30 

9-582840 

9-965615 

9-617224 

10-382776 

9 600700 

9-962398 

9-638302 

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31 

9-583145 

9-965563 

9-617582 

10-382418 

9-600990 

9-96234S 

9-638647 

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29 

3 ‘2 

9-583449 

9-965511 

9-617939 

10-382061 

9-601280 

9-962288 

9-638992 

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28 

33 

9-5S3754 

9-965458 

9-618295 

10-381705 

9 601 570 

9-962233 

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27 

34 

9-584058 

9-965406 

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9-60 IS60 

9-962178 

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26 

35 

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9-964720 

9-623269 

10-376731 

9-605606 

9-961458 

9-644145 

10-355852 

13 

jj  48 

9-588289 

9-964666 

9-623623 

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9-964507 

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10-375317 

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9-625036 

10-374964 

9-607036 

9-961179 

9-645857 

10-354143 

8 

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9-589781 

9-964401 

9*625588 

10-374612 

9 607322 

9-961123 

9-646199 

10-353801 

7 

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9-59008S 

9-964347 

9-625741 

10-374359 

9 607607 

9-961067 

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10-35346C 

6 

9-590387 

9-96429- 

9-626097 

10-373907 

9-607892 

9-961011 

9-646881 

10-353119 

5 

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9-590681 

9-964241 

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9-960955 

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10-352778 

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10-351757 

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01 

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8 9-96402 

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9-609313 

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0 

Cosine 

Sine. 

C otari 

Tang. 

Cosine. 

Sine. 

Cotan. 

Tang. 

67  Deg. 

66  Deg. 

■LOG.  SIXES,  TAX  GENTS,  &e. 


24  Deg:. 

25  Deg. 

' 

| Sine. 

Cosine 

Tang. 

Cotang. 

Sine. 

Cosine. 

Tang. 

Cotang. 

0 

9-009313 

9-960730 

9-648583 

10-351417 

9-625948 

9-957276 

9-668673 

10-331327 

60 

1 

9-609597 

9-960674 

9-648923 

10-351077 

9-626219 

9-957217 

9-669002 

10-330998 

2 

9-609880 

9-960618 

9-649990 

10-350737 

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9-957158 

9 669332 

10-330668 

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3 

9-610164 

9-960561 

9-649602 

10-350398 

9-626760 

9-957099 

9-669661 

10-33033S 

4 

9-610447 

9-960505 

9 649942 

10*350058 

9-627030 

9-957O40 

9-669991 

10-330009 

5 

9-610729 

9-96044. 

9-650281 

10-349719 

9-627300 

9-956981 

9-670320 

10-329680 

6 

9-611012 

9-960392 

9-650620 

10-349380 

9-627570 

9-956921 

9-670649 

10-329351 

54 

7 

9-611294 

9-960335 

9-650959 

10-349041 

9-627840 

9-956862 

9-670977 

10-329023 

53 

8 

9-611576 

9-960279 

9-651297 

10-348703 

9-628109 

9-956803 

9-671306 

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52 

9 

9-611858 

9-960222 

9-651636 

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9-67l635|lO-328365!5i 

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9-612140 

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10-348026 

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9 671963110-328037 

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9-960109 

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10-347688 

9 628916 

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9-672291  10  327709 

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9-652650 

10-347350 

9-629185 

9*956566 

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9-672947!  10-327053 

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9 956447 

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15 

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9-959882 

9 653663 

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9-629989 

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9-673602 

10*326398 

16 

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9-654000 

10-346U0O 

9*630257 

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9 673929- 10-326071 

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25 

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10-342972 

9-632658 

9-955789 

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25 

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9-957979 

9-664703 

10-335297 

9-638720 

9-954396 

9-684324 

10-315676 

12 

49 

9-622956 

9-957921 

9*665035 

10-334965 

9-638981 

9-954335 

9-684646 

10-315354 

11 

j 50 

9-623229 

9-957863 

9-665366 

10-334634 

9-639242 

9-954274 

9-684968 

10-315032 

10 

9-623502 

9-957804 

9-665698 

10-334S02 

9*639503 

9-954213 

9-685290 

10-314710 

9 

9-623774 

9-957746 

9*666029 

10*333971 

9-639764 

9-954152 

9-685612 

10-314388 

S 

53 

9-624047 

9-957687 

9-666360 

10-333640 

9-640024 

9-954090 

9-685934 

10-314066 

54 

9-624319 

9-957628 

9-666691 

10-333309 

9-640284 

9-954029 

9 686255 

10*313745 

6 

55 

9-624591 

9-957570 

9-667021 

10-332979 

9-640544 

9-953968 

9-686577 

10-313423 

5 

56 

9-624863 

9-957511 

9-G67352 

10-332648 

I 9-640804 

9 953906 

9-686898- 10-313102 

4 

57 

9-625135 

9-957452 

9-667682 

10-332318 

I 9-641064 

9-953845 

9-687219  10-312781 

s 

58 

9-625406 

9-957393 

9-66S013 

10-331987 

j 9 641324 

9-95378 3 

9-6S7540 

10-312460 

2 

59 

9-625677 

9-957335 

9-668343 

10-331657 

i 9-641583 

9-953722 

9-687861 

10-31 21 S9 

1 

60 

9-625948 

9-957276 

9-668673 

10-33132' 

; 9-641842 

9-953600 

9-688182 

10-31181S 

0 

Cosine. 

Sine 

Cotan. 

Tang. 

‘ Cosine. 

Sine. 

Cot  an. 

Tang. 

65  Deg. 

64  Deg. 

LOG.  SINES.  TANGENTS,  Lev 


“ °20T 

eg. 

27 

Deg. 

Sine. 

j osine. 

Tang. 

Cotang. 

Sine. 

| Cosine. 

Tang. 

1 Cotang. 

i 

0 

9-641842 

9-953660 

9-688182 

10-311818 

9-657047 

■-’•949881 

9-707166 

'10-292834|60| 

1 

9-642101 

9-953599 

9-688502 

10-311498 

9-65729 

9-949816 

9-707478 

|10- 292522159 

o 

9-642360 

9-953537 

9-6S8S23 

10-311177 

9-65754- 

9-949752 

9-707790 

1 0-2922'  ol  58 

3 

9-64261 S 

9-953473 

9-689143 

10-310857 

9-657791. 

9-949688 

9-7081O2 

10  291S9S 

57 

41  9-642877 

9-953413 

9-689463 

10-310537 

9-658037 

9-94962.3 

9-708414 

10-29I5S1 

561 

5 

9-643135 

9-953352 

9-689783 

10-310217 

9-658284 

9-949558 

9-708726 

10-29  127h 

55 1 

6j  9*643393 

9 953290 

9-690103 

10-309897 

9-658531 

9-949494 

9-700037 

10-29096S 

54 

7 

| 9-643650 

9-953228 

9-690423 

10-309577 

9-658778 

9-949429 

9-709349 

10-290651 

53 

8 

1 9-643908 

9-953166 

9*690742 

10-309258 

9-659025 

9-949364 

9-709660 

10-290.34C 

52 

9 

; 9-644165 

9-953104 

9-691062 

10-308938 

9 659271 

9-949300 

9-709971 

10-290029 

51 

10 

9-644423 

9-95.3042 

9-691381 

10-308619 

9-659517 

9-949235 

9-710282 

10-28971 S 

50 

11 

9-644680 

9-952980 

9-691700 

10-308300 

9-659763 

9-94917U 

9-710593 

10  289407 

49 

12 

9-644936 

9-952913 

9-692019 

10-307981 

9-660U09 

9-949105 

9-710904 

I0-28909C 

4S 

13 

9-645193 

9*952855 

9-692338 

10-307662 

9-660255 

9-949040 

9-711215  10-2SS7S5 

4 r 

U 

9-645450 

9-952793 

9-692656 

10-307344 

9-660501 

9-948975 

9-711525!  10-288475 

46 

15 

9-645706 

9-952731 

9-692975 

10-307025 

9-660746 

9-948910 

9-711836  10-288164 

45 

16 

9-645962 

9-952669 

9-693293 

10-306707 

9-660991 

9-948845 

9-712146 

10-287854 

44 

17 

9-646218 

9-952606 

9-693612 

10-306388 

9-661236 

9-948780 

9-712456 

10-287544 

43 

IS 

9-646474 

9-952544 

9-693930 

10-306070 

9-661481 

9-948715 

9-712766 

10-287234 

42 

19 

9-646729 

9-952481 

9-694248 

10-305752 

9-661726 

9-94S650 

9-713076 

10-286924 

41 

20 

9-646984 

9-952419 

9-694566 

10-305434 

9-66197C 

9-948584 

9-713386 

10-286614 

40 

21 

9-647240 

9-952356 

9-694883 

10-305117 

9-662214 

9-948519 

9-713696 

10-286304 

39 

22 

9-647494 

9-952294 

9-695201 

10-304799 

9-662459 

9-948454 

9-714005 

10-285995 

38 

23 

9-647749 

9-952331 

9-695518 

10-304482 

9-662703 

9-948388 

9*714314 

10-285686 

37 

24 

9-648004 

9-952168 

9-695836 

10-304164 

9-662946 

9-948323 

9-714624' 10-285376 

36 

25 

9-648258 

9-952106 

9-696153 

10-303847 

9-663190 

9-948257 

9-714933)  10-285067 

35 

26 

9-648512 

9.952043 

9-696470 

10-30S530 

9*663435 

9-948192 

9-715242110-284758 

o4 

27 

9-64S766 

9-951980 

9-696787 

10-303213 

9-663677 

9-948126 

9-715551  10-284449 

.33 

28 

9-649020 

9-951917 

9-697103 

10-302897 

9-663920 

9-948060 

9-715S60  10-284140 

;o 

29 

9-649274 

9-951854 

9-697420 

10-302580 

9-664163 

9-947995 

9-7161 681 10-283832 

3 1 

30 

9-649527 

9-951791 

9-697736 

10-302264 

9-664406 

9-947929 

9-716477 

10-283523 

30 

31 

9-649781 

9-951728 

9-698053 

10-301947 

9-66464S 

9-947863 

9-716785 

10-283215 

•29 

32 

9-650034 

9-951665 

9-698369 

10-301631 

9-664891 

9-947797 

9-717093 

10-282907 

28 

33 

9-650287 

9-951602 

9-698685 

10-301315 

9*665133 

9-947731 

9-717401 

10-282599 

2/ 

34 

9-650539 

9-951539 

9-699001 

10-300999 

9-665375 

9-947665 

9-717709 

10-282291 

26 

35 

9-650792 

9-951476 

9-699316 

10-300684 

9 665617 

9-947600 

9-718017 

10-281983 

25 

So 

9-651044 

9-951412 

9-699632 

10-300368 

9-665859 

9-947533 

9-718325 

10-281675 

24 

37 

9-651297 

9-951349 

9-699947 

10-300053 

9-666100 

9-94-7467 

9-718633 

10-281367 

23 

3S 

9-651549 

9-951286 

9-700263 

10-299737 

9-666342 

9-947401 

9-718940 

10-281060 

22 

39 

9-651800 

9-951222 

9-700578 

10-299422 

9-666583 

9-947335 

9-719248 

10-280752 

21 

io 

9-952052 

9*951159 

9-700893 

10-299107 

9-666824 

9-947269 

9-719555 

10-280445 

20 

\{ 

9-652304 

9-951096 

9-701208 

10-298792 

9-667065 

9-947203 

9-719862 

10-280138 

19 

42 

9-652555 

9-951032 

9-701523 

10-298477 

9-667305 

9-947136 

9-720169 

10-279831 

IS 

43 

9 652806 

9-950968 

9-701837 

10-298163 

9-667546 

9-947070 

9-720476 

10-279524 

17 

9-653057 

9-950905 

9-702152 

10-2978481 

9-667786 

9-947004 

9-720783 

10-279217 

16 

43 

9-653308 

9-950841 

9-702466 

10-297534! 

9-668027 

9-946937 

9-721089 

10-278911 

15 

9-653558 

9-950778 

9-702781 

10-297219 

9-66S267 

9-946871 

9-721396 

10-278604 

14 

47 

9-653808 

9-950714 

9-703095 

10-296905 

9-668506 

9-946804 

9-721702 

10-278298 

13 

48 

9-654059 

9-950650 

9-703409 

10-296591 

9-668746 

9-946738 

9-722009 

10-277991 

12 

19 

9-654309 

9*950586 

9-703722 

10-296278 

9-668986 

9-946671 

9-722315 

10-277685 

1 1 

i 50 

9-950522 

9-704039 

10-295964 

9-669225 

9-946604 

9-722621 

10-277379 

10 

9-654808 

9-950485 

9-704350 

10  295650 

9-669464 

9-946538 

9-722927 

10-277073 

9 

52 

9-655058 

9-950394 

9-704663 

10-295337 

9-669703 

9-946471 

9-723232 

10  276768 

8 

53 

9-655307 

9-950330 

9-704976 

10  295024 

9-669942 

9-946404 

9-723538 

10-276462 

7 

54 

9-655556 

9-950266 

9-705290 

10-294710 

9-670181 

9-946337 

9-723844 

10-276156 

6 

55 

9-655805 

9-950202 

9-705603 

10-294397 

9 670419 

9-946270 

9-724149 

10-275851 

5 

50 

9-656054 

9-950138 

9-705916 

10-294084! 

9-670658 

9-946203 

9-724454 

10275546 

4 

9-656302 

9-950074 

9-706228 

10-293772 

9-670896 

9-946136 

9-724760 

10-275240 

3 

5 3 

9-656561 

9-350010 

9-706541 

10-293459 

9-671134 

9-946069 

9-725065 

10-274935 

2 

59 

9-656799 

9-949945 

9 706854 

10-293146 

9-671372 

9-946002 

9-725370 

0-274630 

1 

GO 

9-657047 

9-949881 

9-707166 

10-292834 

9.671609 

9-945935 

9-725674 

L0-274326 

0 

Cosine. 

Sine. 

Cotan. 

Tang. 

Cosine. 

Sine. 

Cotan. 

Tang. 

63  Deg.  62  Deg. 


S52 


LOG.  SINES,  TANGENTS,  kc. 


1 ^^’^^^^^28T>eg~""" 


! -Sine. 

Cosine. 

Tang. 

Cotan  g. 

Sine 

Cosine. 

Tang.  | Coiat.g, 

i i 

9-67160' 

9-945935 

9-725671 

10-274326 

9*685571 

994181c 

9-743752;  10-256248 

60 

9-67184/ 

9-945868 

9-725971 

10-274021 

9-685799 

9 94 1 749 

9-744050  10-255950 

5.4 

9-67208- 

9-945801 

9-726284 

10-273716 

9-686' 127 

9-941679 

9-744318  10-25565-2 

58 

£ 

9-67232 

9-945733 

9-726588 

10-273412 

9-686254 

9-941609 

9*744645]  10*255355 

57 

9-67255S 

9-945666 

9-726895 

10-273108 

9 -68648 

9-941559 

9*744943  10*255057 

56 

£ 

9-672795 

9-945598 

9-727197 

10-272803 

9-684709 

9-941469 

9-745240' 10-254761 

55 

6 

9-67303S 

9-945531 

9-727501 

10-272499 

9-686936 

9-94139! 

9 -745538  1 9-254462 

54 

7 

9-67326S 

9-945464 

9-727805 

10-272195 

9-687167 

9-94132! 

9-745835  10-254165 

53 

3 

9-673505 

9-945396 

9-728109 

10-271891 

9-687389 

9-94125* 

9-746132  10-253868 

52 

9 

9-673741 

9-945328 

9-728411. 

10-271588 

9-687616 

9 94l  is; 

9-74642900-253571 

5i 

10 

9-673977 

9-945261 

9-728716 

10-271284 

9-687847 

9-941 1 1 

9-746726.10-253274 

50 

11 

9-674215 

9-945193 

9-729020 

10-270980 

9-688069 

9-94l04f 

9-747023  10-252977 

49 

12 

9-674448 

9-945125 

9-729323 

10-270677 

9-688295 

9-940975 

9*74731  10*25268 

48 

13 

9-674684 

9-945058 

9-729626 

10-270374 

9-688521 

9-940905 

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47 

14 

9-674919 

9-944990 

9-729929 

10-270071 

9-688747 

9-94o8>4 

9-747913  10-252087 

46 

15 

9-675155 

9-944922 

9-730233 

10-269767 

9-688972 

9-94076.“; 

0-748209  10251791 

45 

16 

9-675390 

9-944854 

9-730535 

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9-68919S 

9-94)69 

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4: 

17 

9-675624 

9-944786 

9-730838 

10-269162 

9-689423 

9-940622 

9-748801110-251199 

4o 

18 

9-675859 

9-944718 

9731141 

10-268859 

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9*94055 1 

9-749097j  10-25090.3 

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19 

9 676094 

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9-749393  10-250607 

41 

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9-944582 

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9-749689  10-250.311 

40 

21 

9-676562 

9-944514 

9-732048 

10-267952 

9-690323 

9-9403.38 

9-749985  10-450015 

39 

2-2 

9-676796 

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9-732351 

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9-690548 

9-940267 

9-750281  10-24971 9]38 

23 

9-677030 

9-944377 

9-732553 

10-267347 

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9 750576,10-249424,37 

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9-677264 

9-944309 

9-732955 

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9-750872!  10-249128 

36 

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9-733257 

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9-678663 

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9-753642  10-247358  30 

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9-752937!  10-247063  29 

32 

9-679128 

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9-735367 

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9-753231 ! 10-246769  28 

33 

9-679360 

9-943693 

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10-264332 

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9-685343 

9-941 S89 

9-743454 

16-256546 

9-698751 

9-93  604 

9-761  l48|l0-iS8S5-| 

1 

60 

9-685571 

9-941819 

9-743752 

10-256248 

9'69S970j 

9-9S7531 

.9-7614S91 10-238561 

0 

Cosine 

Sine. 

Cotan. 

Tang. 

Cosine. 

Sine.  * 

Cota:-.  Tnng.  ' 

61  Peg. 


' icy. 


60  Deg. 


LOCt.  SINES  TANGENTS,  fee. 


"j")  Deg.'* 


8. 

f Cosine 

I Tang. 

Cotan-. 

Sine 

I Cosine.  1 Tan;, 

1 Cotang 

1 

0 9-6US970  9-93753 

1 9-76)43 

9 1 0-23856 

9-7 118391  9-933166  9-778774, 10  221226-6C 

1 9-099189;  9-93715 

8 9-76173 

1 10-23826' 

>71205 

U 9-932990  9-779061 

V -220940  59 

2 9-699407'  9.937.38 

5 9-76202 

3 10-23797 

9-71226 

0 9-932914  9-779346 

10-220654  58 

3 9-699626  9-937,31 

2 9-76231 

4 10-237681 

9-71246 

9 9-9328.38  9-779632 

10-220368  57 

4 9-699844  9-95723 

8 9-76260 

6 10-23739-1 

. 9-.  1267 

9 0-932X2  9-779918 

10-22OU82 

56 

5 9-700062,  9-93716 

5 9-76289 

7 10-23710; 

9-71288 

9 9-9326S 

5,  9-7802  )8 

10-219797 

55 

6 9-700280,  9-9.3709 

2 9-763 IS 

8 1 0-23681 

9-71309 

S 9-9326- 

9,  9-780489  (0-2195 

11 

54 

7 9-700498  9-95701 

9*76347 

9 10-23652! 

9-71330 

8 9-932533!  9-780775 

10-219225 

53 

8 9-700716  9-93604 

9 76377 

1 10-2362.3C 

9-71  351 

7 9-932457  9-781060!  10-2! 894) 

52 

9 9-700933  9-93687- 

9-76406 

10-23593 

9-71372 

3 9-932380!  9 78--^ 

10*21  $65+ 

51 

10  9-701151  9-93679' 

9-76435 

! 10-235648 

9-71393 

5 9-932.304!  9-781631  10-21,8369  .50 

11  9-701368!  9-93672. 

9-76461. 

10-235357 

9-71414 

4 9-93222 

8 9-7S191CO  0-21.3084149 

12  9-70158 

51  9-93665!, 

9-76493; 

10-235067 

9v  1-iJD 

2 9-93215 

1 9-782201 

10-217799  48 

13  9-701  S02l  9-936575 

9-76522- 

10-234776 

9*71456 

9 9.3207 

5 9-782486 

10-217514147 

U 9-70201 

6 9-936505 

9*76551" 

10-234486 

9-/ 1476' 

9-93199 

8 9-782771 

10-217229  46 

15  9-70223 

6 9-9.36431 

9-76580: 

10-234195 

9-71497! 

9-93192 

9-783056 

10  216944' 45 

16  9-70245 

2 9-936357 

9-766095 

10-235005 

9-715181 

9-93184 

9 783341 

10-21 6659' 

44 

17  9-70266 

1 9-9.36284 

9-766385 

10-233615 

9-71539-1 

9 9J176 

9-783626 

10-216374  43. 

18  9-70288 

9-936210 

9-766675 

10-233325 

9*71560:* 

9-95169 

9-78391 01 10-21 6090, 42 

19  9-703101!  9-936136 

9-766965 

10*233055 

9-715803 

9-93161- 

9-784 19s!  l0-215805f4li 

20  9-70331; 

9-936062 

9-767255 

10-232745 

9-716017 

9-93153! 

9-7S4479  10-21552 

1:40 

21  9-70353. 

9-935988 

9*767545 

10-232455 

9-716224 

9 9.5146; 

9-784764  10-315236  39! 

22  9-70.374' 

9-935914 

9-767834 

10-232165 

9-7164.3* 

9-95138. 

9-785048, 10-21495 

381 

23  9-7 0396-1 

9-935840 

9-768124 

10-231876 

9-716639 

9-931306 

9-785.3.32  10-214668  37 

24  9-70417' 

9-935766 

9-768414 

1 0-231 5SC 

9 71G84G 

9-931229 

9-78561 

5 

0 214384 

36 

25  9-704395 

9-935692 

9-768703 

10-231297 

9-71705 3 

9-931152 

9-785900  10-2141 00 

.35 

26  9-7046  a 

9-9S5618 

9-768992 

10-231008 

9-717259 

9-931075 

9-786184  lu-2138  lG‘34 

27  9-704825 

9-935543 

9-769281 

10-230719 

9-717466 

9-950998 

9-786468  10-213532  33 

28  9-705040 

9-935469 

9-769571 

1 0-230429 

9-717673 

9-930921 

9-78675  *110-21 5248 

.32 

29  9-705254 

9-935395 

9-769860 

10-2301  t-i 

9-717879 

9-9.  0845 

9-787036!  10-212964 

31 

30  9-705469 

9-935320 

9-770148 

i 0-229852 

9 718085 

9-930766 

9-78731 9: 10-212681 

.30 

31  9-705683 

9-9.35246 

9-770437 

10-229563 

9-718291 

9-930688 

9'787603|10  212397 

29 

32  9-705898 

9-935171 

9-770726 

10-229274 

9-718497 

9-930611 

9-787886I10-212114 

28 

33  9-706112 

9-935097 

9-771015 

10-228985 

9-7187  >3 

9-9.30533 

U-7SS1 70|  10-211830 

27 

34  9-706326 

9-935022 

9*771.303 

10-2286  7 

9-718909 

9-930456 

9-788453  D-21 1547 

26 

35  9-706539 

9-934948 

9-771592 

10-22840.8 

9-719114 

9-950378 

9-788736}  10-211 264 

25 

56  9-706753 

9-934873 

9-771880 

10-228120 

9 719320 

9-900300 

9-788019  10-21098l|  24 

37  9-706967 

9-9.34798 

9-772168 

10-2 27S5-: 

9 711525 

9-930223 

9-789302}  10-210698|23 

38  9-707180 

9-934723 

9-772457 

10-227545 

9-711730 

9-930145 

9-789585|10  210415-22 

39  9-707393 

9-934649 

9-772745 

10-227255 

9-711935 

9-930067 

9-789868: 10-210132:21 

40  9-707606 

9-934574 

9-773033 

10-226967 

9-7201 40 

9-229989 

9-790151 

10-209849  20 

41  9-707819 

9-93449'' 

9-773321 

10-226679 

9-720345 

9-929911 

9-790434 

10-209566!  19 

42  9-708032 

9-934424 

9-773608 

10-226392 

9-720549 

9-929833 

9-790716 

10-209284;  18 

43  9-708245 

9-934349 

9-773896 

10-226104 

9-720754 

9 929755 

9 790999 

10-209001  17 

44  9-708458 

9-934274 

9-7741 84 

10-225815 

9-72,1958 

9-929677 

9-791281 

10-20871 9!  16 

45  9-708670 

9-934199 

9-774971 

10-225529 

9-721162 

0-929599 

9 791563 

10-208437|15 

46  9-708882 

9-934123 

9-774759 

0-225241 

9-721366 

9 -920521 

9-791846 

10-208154  14 

7 9-709094 

9-934048 

9-775046 

10-224954 

9-721570 

9-929442 

9-792128 

10-207872 

3 

48  9-709306 

9*933973 

9*775333 

10-224667 

9-721774 

9-9-9364 

9-79-2410 

10-207590,12 

49  9-709518 

9-933898 

9 775621 

10-224379 

9-721978 

9-929286 

9-792692 

10-207308  11  | 

50  9-709730 

9-933822 

9-775908 

10-224092 

9-722181 

9 929207 

9-792974 

10-207026,10 

51  9-709941 

9-933747 

9-776195 

0-223805 

9-722385 

9-929129 

9-793256 

10-206744 

52  9-71015S 

9-933671 

9-776482 

0-223518 

9-722588 

9-929050 

9-793538 

10-20646-2 

8 

53  9-710364 

9-933596 

9-776769 

0-2232  >2 

9-722791 

9-928972 

9-793819 

10  206181 

7 

54  9-710575 

9-933520 

9-777055 

0 222945 

9-722994 

9-928853 

9-794101 

10-205899; 

55  9-710786 

9-933445 

9-777342 

0-222658 

9-723197 

9-928815 

9-794383 

10-205617 

£ 

56  9-710997 

9-933369 

9-777628 

0-222372 

9-723400 

9-928736 

9-79  4664 

10-205336 

4 

57  9-71 1 208 

9-933294 

9-777915 

■ -222085 

9-723G03 

9-928657 

9794946 

10-205054 

58  9-71  419 

9-933217 

9-778201 

0-221799 

9*723805 

9-928578 

9 795227 

10-204773 

5 

56  9-711629 

9-933141 

9-778488 

0 221512 

9-724007 

9-928499 

9-795508 

10-204492 

60!  9-711839 

9-933066 

9-778774 

10-221226  : 

9-724210 

9-928420 

9-795789 

10  204211 

0 

Cosine. 

Sine. 

Cotan. 

Tang.  ] 

Cosine. 

Sine.  1 

Cotan. 

Tang. 

59  Deg- 

58  Deg. 

5 


LOG.  SINES,  TANGENTS,  fcc. 


32  Deg. 

S3 

eg- 

Sine. 

Cosine.  | 

Tang,  j 

Cotang. 

Sine.  ( 

Cosine.  ( 

Tang.  | 

Cotang.  I 

'o 

9-724210 

9-92842(4 

9-795789  1 

0-204211 

9-736109 

9-923591 

9-812517(10-18.'  87,! 60 

1 

9-724412 

9-928342 

9-796070  l 

u-203930 

9-736303 

9-923509 

9-512794  10-187206  59 

2 

9724614 

9-928263 

9-796351  l 

0-203649 

9-736498 

9-923427 

9-813070  10-186950  58 

3 

9-724816 

9-928183 

9-796632  l 

0-203368 

9-736592 

9*923345  i 

9-8133471 10-186653  57 

4 

9-725017 

9-928 104 

9-796913 

0-203087 

9 736886 

9-923263, 

9-813625  10-186377  56 

5 

9-725219 

9-9280251 

9-797194 

0-202806 

9-737080 

9-920131 

9-813899  10-186101,55 

6 

9-725420 

9-927946 

9-797474 

0-202526 

9 737274 

9-925098 

9-814176 

10-1858241 

Si 

7 

9-725622 

9 927867 

9-797755 

10-202245 

9-737467 

9-92301  g! 

9-814452 

10-185543l53i 

8 

9-725823 

9-927787 

9-/ 98036  10-201964 

9-737661 

9-922933! 

9-814728 

10185272 

62a 

9 

9-726024 

9-927708 

0/93316 

10-201684 

9-737855 

9-922851; 

9-815004 

10-184996151 1 

10 

9-726225 

9-927629 

9-798596 

10-201404 

9-738048 

9-922768 

9-815280!  10-184720 

50 

11 

9-726426 

9-927549 

9-798877 

10-201123 

9-738241 

9-922686 

9-815555110  184445  49 

12 

9-726626 

9 927470 

9-799157 

10-200843 

9-738434 

9-922603 

9-81583lltO-184l69:48 

13 

9-726827 

9-927390 

9-799437 

10-200563 

9-738627 

9-922520 

9-816107 

10-183893S47 

14 

9 727027 

9-927310 

9-799717 

10-200283 

9-738820 

9-922438 

9 8)6382  10  183618'46 

15 

9-727228 

9-927231 

9-799997 

10-200003 

9-739013 

9-922355 

9-8 16658!  10-183342 

**5 

IB 

9-727428 

9-927151 

9-800277 

10-199723 

9.739206 

9-922272 

9-8l6933|  10-183067 

44 

17 

9-727628 

9-927071 

9-800557 

10-199445 

9-739398 

9-922189 

9-817209:10-182791  43 

18 

9-727828 

9-926991 

9-800836 

10-199164 

9-739590 

9-922106 

9-8174S4jlO-182516 

4 2 

19 

9-728027 

9-926911 

9-801116 

10-198884 

9-739783 

9-922023 

9817759  10-182241 

41 

20 

9-728227 

9-926831 

9-801396 

10-198604 

9-7S9975 

9-921940 

9-818035  10-181965 

40 

21 

9-728427 

9-926751 

9-801675 

10-198325 

9-740167 

9-921857 

9-818310.10-1 81690 

39 

22 

9-728626 

9-926671 

9-801955 

10-198045 

9-740359 

9-921774 

9-8l8585jlO-181415 

38 

23 

9-728825 

9-926591 

9-802234 

10-197766 

9-740550 

9-921691 

9-8lS860|l0-l8ll40 

57 

24 

9-729024 

9-926511 

9-802513 

10-197487 

9-740742 

9-921607 

9-819135I10-1S0S65 

-'6 

25 

9-729223 

9-926431 

9-802792 

10-197208 

9-740934 

9-921524 

9-8194lO[t0180590 

35 

26 

9-729422 

9-926351 

9-803072 

10-196928 

9-741125 

9-921441 

9-819684  10-180316 

34 

27 

9-729621 

9-926270 

9-803351 

10-196649 

9-741316 

9-921357 

9-8l9959[l0-18004l 

13 

28 

9-729820 

9 926190 

9-803630 

10-196370 

9-741508 

9-921274 

9-8202341 10  179766 

- 

29 

9-730018 

9-926110 

9-803909 

10-196091 

9-741699  9-921190 

9-820508  10-1 79492 

31 

30 

9-730217 

9-926029 

9-804187 

10-195813 

9-741889 

9-921107 

9-820783 

10  179217 

30 

31 

9-730415 

9-925949 

9-804466 

10-195534 

9-742080 

9-921023 

9-821057 

10-17S943 

29 

32 

9-730613 

9-925868 

9-804745 

10-195255 

9-742271 

9-920939 

9-821332 

10-178668 

28 

33 

9-730811 

9-925788 

9-805023 

10-194977 

9-742462 

9-920856 

9-821606 

10-178394 

27 

34 

9-731009 

9-925707 

9-S05302 

10-194698 

9-742652 

9-920772 

9-821880 

10-178120 

25 

35 

9-731203 

9-925 6QC 

9-805580 

10-194420 

9-742842 

9-920688 

9-822154 

10-177846 

25 

3fi 

9-731404 

9-925545 

9-805859 

10-194141 

9-743033 

9-920604 

9-822429 

10177571 

.4 

37 

9-731602 

9-925465 

9-806137 

10-193863 

9-7-13-223 

9-920520 

9-822705 

10-177297 

25 

38 

9-731799 

9-925384 

9-806415 

10-193585 

9-74S413 

9-920436 

9-822977 

1017702 3 

22 

39 

9-731996 

9-925303 

9-806693 

10-193307 

9-743602 

9-920352 

9-823251 

10-176749 

21 

40 

9-732193 

9-925222 

9-806971 

10193029 

9-743792 

9-920268 

9-823524 

10-176476 

- 

41 

9-73239C 

9-925141 

9-807249 

10-192751 

9-743982 

9-920184 

9-823798 

10-176202 

19 

42 

9-732587 

9-925060 

9*80/521 

10-192473 

9-744171 

9-920099 

9-824072 

10-175928 

18 

43 

9-732784 

9-924979 

9-807805 

10-192195 

9-744S61 

9-920015 

9-824345 

10-175655 

17 

44 

9-732980 

9-924897 

9-8080S3 

10-191917 

9-744550 

9-919931 

9-824619 

10-175381  1 6 

45 

9-733177 

9-924816 

9-S08361 

10-191639 

9-744739 

9-919846 

9-824893 

10-175107  15 

46 

9-733373 

9-924735 

9-808638 

10191362 

9-744928 

9-919762 

9-825166 

10-174834  14 

47 

9-733569 

9-924654 

9-808916 

10-191084 

9-745117 

9-919677 

9-S25439 

10-174561  13 

48 

9-733765 

9-924572 

9S09193 

10T90S07 

9-745506 

9-919593 

9-82571S 

10T74287 

12 

49 

9-733961 

9-924491 

9-809471 

10-190529 

9745494 

9-919508 

9-825986 

10-174014 

11 

50 

9-734157 

9-92440! 

9-809748 

10190252 

9-745683 

9-919424 

9-S26259 

10-173741  10 

51 

9-734355 

9-92432S 

9-810025 

10-189975 

9-745871 

9-919339 

9-826532 

10-173468 

9 

52 

9-734549 

9-924246 

9-810302 

10-189698 

9-746060 

9-919254 

9-826805  10-173195 

8 

53 

9-734744 

9-924164 

9-810580 

10-189420 

9 746248 

9-919169 

9-S27078 

10-172922 

7 

54 

9-734939 

9-924086 

9-810857 

10-189143 

9-746436 

9-919085 

9-827351 

10-172649 

6 

55 

9-735135 

9-924001 

9-811134 

10-188866 

9-746624 

9-919000 

9-827624 

10-172S76 

5 

56 

9-735330 

9-923919 

9-811410 

10-188590 

9-746812 

9-918915 

9-827897(10-172103 

4 

57 

9 73552: 

9-923837 

9-811687 

10-188313 

9-746999 

9-918SS0 

9-8281 7010-171SS0, 

s 

58 

9-735719 

9-923755 

9-811964 

10-188036 

9-747187 

9-918745 

9-828442 

10-171558 

O 

59 

9-7359G 

9923676 

9-812241 

10-187759 

9-747374 

9-918659 

9-82S715  10-171285 

i 

60 

9-736109 

9-923591,  9-812517 

10187483 

9-747562 

9-918574 

9-82S987 

10-17I01S| 

0 

Cosine. 

Sine. 

1 Cotan. 

Tang. 

Cosine. 

Sine. 

Cotan. 

Tang.  1 

57  Deg. 

56  De_ . 

tt;  fcfcfegggj  asggjgas 


LOG.  SINES,  TANGENTS,  &c, 

S^BegT 


a 9-74849; 


9 74961. 


9-759944 

9-753128 

9 753312 
9-753495 
9-753679 
9-753862 
9-754046 
9-754229 

-9-754412 

9-754595 

9-754778; 

9-754960 

9-755143 

9-755326; 

9-755508 

9-755690 

9-755872 

9-756054 

9-756236 

9-756418 

9-756600 

9-756782 

9-756963 

9-757144 

9-757326 

9-757507 

9-757688 

9-757869 

9-758050 

9-75S230 

9-758411 

9-758591 


Cosine. 


Cosine 

Tang. 

Cotang. 

a 9-91  8574 

9-S2S9S7 

10-171013 

.1  9-918h89 

9-829261 

10-170740 

6 9-918404 

9-829532 

10' 170468 

3 9-918318 

9-829805 

10-170195 

’ 9*9] 

9-830077 

10-169923 

9-918147 

9 830349 

10-169651 

9-918062 

9-830621 

10169379 

9-917976 

9-830893 

10-169107 

9-9 : 7S91 

9-831165 

1016S835 

9-917S05 

9-831437 

10-168563 

9-917719 

9-S317U9 

10-168291 

9-917634 

9-831981 

10-168019 

9-917548 

9-832253 

1016774 7 

9-917462 

9-832525 

10-167475 

9-917376 

9-832796 

10-167204 

9-917290 

9-83306S 

10  166932 

9 917204 

9-833839 

10-160661 

9-9171  IS 

9-833611 HO-166389 

9-917032 

9-S338S21 10-166118 

9-916946 

9-834154  10-165846 

9-916S59 

g-834425!  10-165575 

9-916773 

9-834696 1 10-165304 

9-916687 

9-834967 

10-165033 

9-916600 

9-835238 

10-164762 

9-916514 

9-835509 

10-164491 

9-916427 

9-835780 

10-164220 

9-916341 

9-S36051 

10-163949 

9-916254 

9-836322 

10-163678 

9-916167 

9-836593 

10-163407 

9-916081 

9-836864 

10-163136 

9-915994 

9-837134 

10-162866 

9-915907 

9-837405 

10- 162595 

9-915820 

9-837675 

10-162325 

9-915733 

9-837946 

10-162054 

9-915646 

9-838210 

10-161784 

9-915559 

9-838487 

10-161513 

9-915472 

9-83875 7 

10-161243 

9-9153S5 

9-S39027 

10-160973 

9-915297 

9-839297 

10-1 60703 

9-915210 

9-83956S 

10-160432 

9-915123 

9-839838 :10-1601 62 

9-915035 

9-840108  10-159892 

9-914948 

9- 840378 1 10-159622 

9-914860 

9-840648)10-159352 

9-914773 

9-840917 

10-159083 

9-914685 

9-841 187!  10-15881 3 

9-914598 

9-841457  i 10-158543 

9-914510 

9-841 727;  10-158273 

9-914422 

9-841996,10-158004 

9-914334 

9-S42266' 10-157734 

9-914246 

9-842535  10-157465 

9-914158 

9-842805)  10-157195 

9-914070 

9-843074!  10-156926 

9-913982 

9-843343!  10-156657 

9-913894 

9-843612)  10-156388 

9-913S06 

9-S43882’lO-156ll8 

9-913718 

9-844151 

10-155849 

9-913630 

9-844420 

10-155580 

9-913541 

9-844689 

10-155511 

9-913453 

9-844958 

10-155042 

9-913365 

9-845227 

10-154773 

Sine 

Cotan. 

Tang. 

Sine.  I Cosine.  I Tang,  j Cotang 


9-758591 

9-758772 

9-758952 

9-759132 

9-759312 

9-759492 

9-759672, 

9-759852 
9-760031 
9 760211 
9-760390 
9 760569 
9-760748 

9-760927 

9-761106 

9-761285 

9-761464 

9-761642 

9-761821 

9-761999 

9-762177 

9’762356 

9-762534 

9-762712 

9-762889 

9-763067 
9 763245 
9-763422 
9-763600 
9-763777 
9-763954 

9-764131 

9-764308 

9-764485 

9-764662 

9-764838 

9-76501 

9-765191 

9-765367 

9-765544 

9-765720 

9-765896 

9-766072 

9-766247 

9-766423 

9-760598 

9-766774 

9-766949 

9-767124 

9-767300 

9-767475 

9-767649 

9-767824 

9-767999 

9-768173 

9-768348 

9-768522 

9-768697 

9768871 

9-769045 

9-769219 

Cosine. 


-I- 


9-913365 

9-913279 

9-91318 

9 913099 

9-913010 

9-912922 

9-912833 

9-912744 

9-912655 

9-912566 

9-912477 

9-912388 

9-912299 

9-912210 
9 91212] 
9-912031 
9-911942 
9-911853 
9 911763 

9-911674 
9-911584 
9 911495 
9-911405 
9-911315 
9-911226 
9-911136 
9-911046 
9-910956 
9-910866 
9-910776 
9 910686 

9-910596 
9-910506 
9-910415 
9 910325 
9-910235 

9-910144 

9-910054 

9-909963 

9-909873 

9-909782 

9-909691 

9-909601 

9-909510 

9-909419 

9-909328 

9-909237 

9-909146 

9-909055 

9-908964 

9-D08873 

9-908781 

9-908690 

9-908599 

9-908507 

9-908416 
9 908324 
9-508233 
9-908141 
9-908049 
9-907958 


Sine. 


9-845227 110-1 54/  73 
9-8454961 10-154504 
9 845764)  10-154256 
9-846036  10-153967 
9-S46302;  10-153698 
9-846570;  10-1 53430 
9-S46S39, 10-153161 

9-847108!  10-152892 
9-8473761 10-152624 
9-847644  10-152356 
9-8479131 10-152087 
9 848181  10  15J  819 
9-848449  10-151551 


9-848717 
9-848986 
9-849254 
9 849522 
9-849790 
9-850057 


10-151283 

10151014 

10-150746 

10-150478 

10-150210 

10-149943 


9-850325,10-149675 
9-850593!  10-149407 
9-850861!  10-149139 
9-85 1 1 29|  1 0-1 4887 ! j38 
9-851596:10-14S6u4  37 
9-851664  10-148336  36 


9-851931 
9-852199 
9 852466 
9-852733 
9-853001 
9 853268 


10-148069 
10-147801 
10'l47534 
10-147267 
10-146999 
10-146732 

9-853535  10-146465  29 
9-853802  10-146l9s;28 
9-854069!  10-145931  27 
9-854336  10-145664 
9-854603  10-145397 
g-854870  10-145130 

9-85513710-144863 
g-855404ll0-144596  - 

g-So5671 j 10  144329 12 . 
9-855938!  10-144062)20 
9-856204:10-143796  39 
9-856471 1 10-143529- 18 

9 856737  10-1 -i 3263!  17 
9-857004!l0-142996;i6 
9-85727010-142730)15 
9-857537|10-l42463  14 
9 857803  10-142197 
9-S5S069  10-141931 


9-858336 

9-858602 

9-858S68 


13 
12 

10-141664  11 
10-141398  10 
10-141132)  9 
9-859134)10-140866)  8 
9-859400110-140600  7 
9-859666)  10-143340 j 6 
9-859932  10-140068;  5 
9-8C019S,  10-1 39802  4 
9-S60464!  10-139536)  3 
9-860730  10-139270 
9-860995 
9-861261 
’otan. 


55 


£5 


54  Deg. 


10-139005 1 1 
10-138739'^0 

TlniTl  | 


LOG.  SINES,  TANGENTS,  ke. 


36  Deg. 37  ‘’beg. 


Sine.  | 

Cosine.  | 

Tang,  j Coiaug. 

bine,  j 

• osine. . | 

Tan-  j 

Cota  ng. 

0 

9-709219 

9-907958 

9-861261  ■ 10-138739 

9*77946.- 

9-902349) 

9-8771 N 

10.122886  60 

1 

9-769393 

9-9078G6 

9-86l527|l  0 1 38473 

9-7796 31 1 

'-’•902253 

9*877377 

10.122623159 

2 

9-769566 

9-907774 

9-861 792  I O'l  .,8208 

9-779798; 

9-902158 

9*8  77640 

10-|  22360  58 

; 3 

9-769740 

9-907682 

9*862058!  10-137942 

9-779D66| 

9-902063 

9-877903 

10-122097.57 

4 

9-7C9913 

9-607590 

9-862323!  1 0-137  677 

9-780133 

9-901967 

P-87816- 

10-121835 

56 

5 

9-770087 

9-907498 

9*662589  10-I374i  1 

y-766300 

9-901872 

9-S784-d8 

10*121572 

55 

: '6 

9-770260 

9 907406 

9-862854  10-137146 

9-780467 

9-901 776 

9-878691 

10*121309 

54 

7 

9*770433 

9-907314 

9-863119  10-136881 

9-780634 

9-901681 

9-878953 

10*1210-17 

- 

8 

9-770606 

9-907222 

9-863385  10-156615 

9-780801 

9-901585 

9-879216 

10*120784 

5 . 

1 9 

9-770779 

9-907129 

9-865650  10-136350 

9-780y68 

9-901490 

9-879478 

10-120522 

10 

9-770952 

9-907037 

9-863915  10-136085 

9-781134 

9-501394 

9 87  -741 

10 120259 

n 

9-77 11  2d 

9-906945 

9-864180  10-135,820 

9-781301 

9-901298 

9-8800U3 

10-119997 

12 

9-771298 

9-906852 

9-8644451 10-135555 

9-781468 

9-901202 

9-880265 

10-119755 

48 

13 

9-771470 

9-906760 

9-86-;710'10'135290 

9-781634 

9-901106 

9-880528 

10-119472 

14 

9-771643 

9 906667 

9*8649751 10-135025 

9-731800 

9-901010 

9-880720 

10119210 

46 

15 

9-771815 

9-906575 

9-865240110-1 84760 

9*781966 

9-90u9l4 

9-88105*2 

10-1 18948145 

16 

9-771987 

9-906482 

9-865^05  10-184495 

9*782132 

9-90081 8 

9-881314 

10-1  IS686'44 

17 

9-772159 

9-906389 

9-865770;  10-1 34230 

9-782298 

9-900722 

9-881577 

10-118423 

4-^ 

IS 

9-772331 

9-906296 

9-866035  10-133965 

9-782464 

9-90062C 

9-8S1859 

10-llSlGl 

42 

ID 

9-772503 

9-906204 

9-866300  10-13370O 

9-782630 

9-900529 

9S82101  10-117899 

il 

20 

9*7726/5 

9-906111 

P-8065641 10-133436 

9-782796 

P-9004.53 

9*88-2363  JOT  17637 

40 

21 

9-772847 

9-606018 

9-866829|10*l3317l 

9-782961 

9*900337 

9-88*2625 

10-117.575 

59 

1 22 

9-77301 8 

9-905925 

9-86/094  1 0-1 32906 

9-783127 

9-900240 

9 8 1*2887 

10*117113 

.38 

23 

9-773)90 

9-905832 

9-867358!  10-132642 

9-783292 

9-900144 

9*883148  10*116852 

” 

24 

9-773361 

9-905739 

0-867623ll0*l  32377 

9-783458 

9-900047 

9-S-3410 

10-116590 

36 

25 

9 ’773533 

9-905645 

9*867887j  10*1 321 13 

9-7S3623 

9S99951 

9*8S3672|l0116328 

35 

26 

9*773704 

9-905552 

9-868152  10-131848 

9-783788 

9-899854 

9*8i3934|  1 0*1 1 6066 

34 

27 

9*/  7 387  5 

9-905459 

9-868416!  .0-131584 

9-783953 

9-899757 

9*884:96  10115804 

33 

9-774o46 

9-905366 

9*8686SO|  1 0-1 .>1320 

9-784118 

9-SP9660 

9 854457 

10-11554.) 

■>*_ 

29 

9-774217 

9 905272 

9*868945 1 1U-1 .41055 

9-784282 

9-899564 

9*884719|KH  15281 

,31 

30 

9-774388 

9 905179 

9-869209,10*1 3U791 

9-784447 

9-899467 

9-884980  lO'll  5U20  30 

31 

9-774558 

9-905085 

9-869473  10-130527 

9-784612 

9-899370 

9-885-24240  1 14758  29 

32 

9-774729 

9-904992 

9-8697S7i  10-130263 

9-784777 

9-899-273 

9*885504  10-114496  28 

35 

9-774899 

9-904898 

9-870001  10-129999 

9-784941 

9-899176 

9-835765i10-l  142.55 

54 

9-775070 

9-9O4804 

9-8702651 10-129735 

9-785105 

9-895078 

9-886026  In- 113974  ‘26 

35 

9-775240 

9-904711 

D-S70529;  10-129471 

9-7S5269 

9-S98981 

9-8S6-28S  10-113712 

1-25 

i36 

9-775410 

9-904617 

9*870793 10*139207 

9-785433 

9-898884 

9 886549  10-113451  24 

[ 37 

9-775580 

9-904523 

9-871057!  10*128943 

9-785597 

9-898787 

9-SS.5811 

10-113189  23 

■38 

9-775759 

9-904429 

9*871 321 ! 1 0*1 28679 

9-785761 

9-898689 

9-887072  10112928  22 

39 

9-775920 

9-904335 

9*871 585!  10*12841 5 

9-785925 

9-898592 

9 887.335 

10-11-2667 

2! 

40 

9-776090 

9-904241 

9-871849! : 0*128151 

9-7S60S9 

9-898494 

9 887594  10-112400 

■20 

41 

9-776-259 

9-904147 

9-872112  10-127888 

9-786252 

9-898597 

9-887855  10-1 12145 

IP 

42 

9-776429 

9-904053 

9-872370 11 0*1 27624 

9-786416 

9-898299 

9*8S8116  101118S4 

IS 

: 43 

9-776598 

9-90395  ’ 

9-S72640j  10-127360 

9-786579 

9 898202 

9*888378  10-111622 

17 

44 

9-776768 

9-9U3S64 

9-872903(10-127097 

9-786742 

9-89S104 

9-8886.59(1 0-1 11 361 

16 

45 

9-776937 

9-903770 

9873l67|  10*126833 

9-786906'  9-S98006 

9-888900 

'10-111100 

15 

,46 

9-777106 

9-903676 

9-873430;  1 0-126570 

9-7S7069 

9897908 

9-889161 

10-110839 

14 

47 

9-777275 

9-903581 

9-873694, 10-126306 

9-787232 

9-897810 

9 8 8942 1 

10-1 10579 

15 

48 

. 9-777444 

9-903487 

9-873957,10-126043 

9-787395 

9-897712 

9-SS9682 

10110318 

12 

49 

9-777613 

9-903392 

9-S74220ll0-1257S0 

9-7S7557 

9-S97614 

9-889943  10110057 

11 

50 

9-777781 

9-903398 

9-S74484  10*125516 

9-787720 

9-897516 

9-89  -204.10-109796 

10 

51 

9-777950 

9-G03203 

9-874747  10-125253 

9-7S7383 

9-S97418 

9-890465  i 1 0-109535 

9 

52 

9-778119 

9-903108 

9-875010,10-124090 

9-788045 

9-897.320 

9-890725  !l  0-1 09275 

S 

53 

9-778287 

9-903014 

9-875273  1 0-1  %4727 

9-7S820S 

9-897222 

9-890986  10-109014 

r 

54 

9-778455 

9-902919 

9-875537  10-124663 

9-788370 

9-897123 

9-891-247 

10-10875S 

6 

55 

9-778624 

9-902824 

9-S75800, 10-1 24200 

9-7SS532 

9-S97025 

9-S91507 ' 10-108493 

5 

56 

9-778792 

9-902729 

9-876063' 10*12393 

9 7SS694 

9-S96926 

9-89.768 

10-10S2S2 

4 

57 

9-778960 

9-902634 

9-S76826  10  123674 

9-7S8S56 

9-896828 

9-892028 

10*107972 

3 

58 

9-779128 

9-902539 

9-8765S9]  10*123411 

9-78901 8 

9-S96729 

9-892289 

10*10771  lj 

0 

50 

9-779295 

9-902444 

9-S76852  10-123148 

9-789180 

9-896631 

9-89254P 

10*107451 1 

1 

60 

9*779463 

9*902349 

9-S77U4I10-122886 

9-789342 

9-S96532 

9-S92S10 

1 0*1 071  POj 

« 

3 

Cosine. 

Sine. 

Cotan.  1 Tang. 

Cosine.  I 

Sine. 

Cotan. 

Tang.  1 

LOG.  SINES,  TANGENTS,  See; 


38  lie 

39  Uog. 

Sine.  | 

Cosine. 

Tang. 

Cotang,  f 

Sine,  j 

Cosine.  ] 

Tang. 

Co  tang. 

0 

9-789342 

9-896532 

9-892810 

10-107190 

9-79887-1 

9-8905031 

9-9u8369 

10091631  60 

1 

9-789504 

9-S96433 

9-393070 

10-106930- 

9-799o28 

9-89n400 

9-9081,28 

10-091372 

59 

2 

9-789665 

9-896336 

9 -893331 

10-106669 

9-799184 

9-890298 

9-908886 

iu  u9ni4 

58 

5 

9-7S9827 

9-896236 

9-893591 

10-1064-09 

9-799339 

9-8900J5 

9-909i  44 

10-090856 

57 

4 

9-7899SS 

9-896137 

9-893S51 

10-1061491 

9-799495 

9-890093 

9-909402 

i 0-090598 

56 

5 

9-790149 

9-896038 

9-894111 

10-105889! 

9-799651 

9-889990 

9-9o9660 

10-090340  55 

6 

9-790310 

9-895939 

9-894372 

10-105628 

9-799806 

9 889888 

9-909918 

10-090082:54 

7 

9-790471 

9-895840 

9-894632 

10-105368 

9-799962 

9-889785 

9-910177 

10-089823'53 

8 

9 790632 

9-895741 

9-894891 

10-105108 

9-8001 1 7 

9-839682 

9-910435 

10-089565 

52 

9 

9-790793 

9-895641 

9-895152 

10-104848 

9-800222 

9-889579 

9 9U/693 

10-089307 

51 

10 

9-790954 

9-895542 

9*595412 

10-104588 

9-800477 

9-889477 

•9-910951 

10-089049 

50 

11 

9-791115 

9-895443 

9-895672 

10-104328 

9-800582 

9-889374 

9-91 12U<- 

10  088791 

49 

12 

9-791275 

9-895343 

9-895932 

10-104068 

9-8U0737 

9-889271 

9 9/ 1 467 

10-088533148 

13 

9-791436 

9-895244 

9-896192 

10-103808 

9-800892 

9 889168 

9-911725 

10-088275  47 

14 

9-791596 

9-895145 

9-896452 

10-103548 

9-SU1U47 

9-889064 

9-911982 

10-088018 

46 

15 

9-791757 

9-895045 

9-896712 

10-103288 

9-801201 

9-888961 

9-912240 

10-087760  45 

10 

9-791917 

9-894945 

9-896971 

10-103029 

9-801356 

9-SS8S58 

9-912498 

10-087502 

44 

17 

9-792077 

9-894846 

9-897231 

10-102769 

9-801511 

9-888755 

9-912756 

10-087244- 

43 

18 

9-792237 

9-894745 

9-897491 

10-102509 

9-801665 

9-888651 

9-913014 

10-086986 

42 

3 10 

9-792397 

9-894646 

9-897751 

10-102249 

9-801819 

9-888548 

9-913271 

10-086729 

41 

120 

9-792557 

9-894546 

9-89S01U 

10-101990 

9-801973 

9-888444 

9-913529 

10-086471  40 

121 

9-792716 

9-894446 

9-89S270 

10-101730 

9-80212- 

9-883341 

9-913787 

10  086213  39 

122 

9-792876 

9-894346 

9-898530 

10-101470 

9 802282 

9-8882.37 

9-914044 

10-08595g!s8 

23 

9-793035 

9-894246 

9-898789 

10101211 

9-802436 

9-888134 

9-914302 

! 0-1)85698  37 

24 

9-793195 

9-894146 

9-899'  49 

10-100951 

9-802589 

9-888030 

9-914560 

10-085440  36 

25 

9-793354 

9-894046 

9-899308 

10-100692 

9-802743 

9-887926 

9-914S17 

10-085183  35 

26 

9-793514 

9-893946 

9-899568 

10-100432 

9-802897 

9-887823 

9-915075 

lU-084925'34 

27 

g-793673 

9-893846 

9-899827 

.0-100173 

9-803050 

9-887718 

9-9)5332 

100846681.53 

28 

9-793832 

9-893745 

9-900087 

10-0999)3 

9-803204 

9-8S7614 

9-915590 

10-084410  32 

29 

9-793991 

9-893645 

•9-900346 

10-099654 

9;803357 

9-8875U) 

9-915847 

10-084153,51 

30 

9-794150 

9-893544 

9-900605 

10-099395 

9-8035  il 

9-8874  06 

9-916 IU4 

10-083896!  30 

31 

9-794308 

9-S93444 

9-900804!  10-099136 

9-803664 

9-887302 

9-916362 

10-083638  29 

32 

9-794467 

9-893343 

9-901 124!l  0-098876 

9-803817 

9-88719S 

9-9  6619 

10-0S3381  28 

33 

9 794626 

9-893243 

9-901383:10-098617 

9-803970 

9*887093 

9-916877 

10-083123|27 

1 34 

9-794784 

9-893142 

9-901642‘lO-09i358 

9-804123 

9-886989 

9-9 1 71 34 

1O-082S66!  26 

| 35 

9-794942 

9-893041 

9-901901  10-098099 

9-804276 

9-S86S85 

9-917391 

10-OS26O9 

25 

| 36 

9-795101 

9 S92940 

9-902165  10-097840 

9-804428 

9-886780 

9-917648 

10-082552  j 24 

fj  37 

9-795259 

9-892839 

9-902420  10-097580 

9-804581 

9-886676 

9-917906 

10,082094|23 

i 38 

9-795417 

9-892739 

9-9U2679 

10-097321 

| 9-S04734 

9-8865/ 1 

9-918163 

(0-081837  22 

| 39 

9-795575 

9-892638 

9-90293S 

10-097062 

i 9-804886 

9-886466 

9-918420 

10-081580 

21 

1 40 

9-795733 

9-892536 

9-903197 

10  096803 

| 9-805039 

9-886362 

9-918677 

10-081323 

20 

41 

9-795891 

9-892432 

9-903456 

10-096544 

j 9-805191 

9-88625  i 

9-918934 

10-081066 

19 

42 

9-796049 

9-892334 

G-90S714 

10-096286 

| 9-805343 

9-886152 

9-Si9l9l 

10-080809 

18 

43 

9-796206 

9-892233 

9-903973 

10-096027 

j 9-S05495 

9-S86047 

9-919448 

10*0£0552 

17 

44 

9796364 

9-892132 

9-904232 

10-095768 

9-805647 

9-885942 

9-919705 

10-080295 

16 

45 

9-796521 

9-S92030 

9-904491 

10-095509 

; 9-805799 

9-885337 

9-91 9962 

10-080038  15 

j 46 

9-796679 

9-891929 

9-904750 

10-095250 

1 9-805951 

9-885732 

9-920219 

10-079781 

14 

47 

9-796836 

9-891827 

9-905008 

10-094992 

1 9-806103 

9-885627 

9-920476 

! 10-079524 

IS1 

48 

9-796993 

9-891726 

9-905267 

10094733 

; 9-806253 

9-885522 

9-920733 

j 10-079267 

12 

!49 

9-797150 

9-891624 

9-905526 

10  094474 

j 9-806406 

9-885416 

9-920390 

il  0 079710 

11 

50 

9-797307 

9-891523 

9-905785 

10-094215 

9-306557 

9-88531  i 

9-921247 

llC-078753  111 

51 

9-797464 

9-891421 

9-906043 

10-093957 

I 9-806709 

9-885205 

9-921503 

! 10  078497 

9 

52 

9-797621 

9-891319 

9-906302 

10-09369S 

! 9 S06860 

9-885100 

9-921760 

. 0-078240 

1 8 

53 

9-797777 

9-891217 

9-906560 

10  093440 

! 9-807111 

9-884994 

9-922017 

10-077983 

/ 

54 

9-797934 

9-801115 

9-906819 

10-093181 

| 9-S07163 

9-S84889 

9-922274 

10-077726  ; 6 

55 

9-798091 

9-891013 

y-907077 

10-092923 

! 9-807314 

9-8S4783 

9-922530 

10-077470 

5 

56 

9-798247 

9-89091 1 

9-907336 

1O-092G64 

| 9-807465 

9-884677 

9-922784 

10-077213 

4 

57 

9-798403 

9 -89081 '9 

9-907594 

10-092406 

1 9-80.615 

9-884572 

9-9230S4 

10-076956 

3 

58 

9-79856H 

- 9-890707 

9-907853 

10092147 

j 9-807766 

9-884466 

9-923300 

10-076700 

2 

59 

9-798716 

9-890605 

9-908111 

10-091889 

9-8079171  9-884360 

9-923557 

10076443 

60 

9-798872 

9-890503 

9-908369 

10-091631 

j 9-S0S067 

9-884254 

9-923814 

10-076186 

o| 

Cosine. 

Sine. 

Cotan. 

Tang. 

| Cosine. 

Sine. 

Ce  l an. 

Tang. 

j 

51  Deg. 

50 

fiSK 

SdJ  1 

LOG.  SINES,  TANGENTS,  A c, 


| . Sine. 

Cosine. 

Tang 

Colang.  j 

Sine. 

Cosine. 

fang.  | Count. 

— 

S 0 

9-808067 

9-884254 

9-923814, 10-07 31 86 1 

9-810943 

9-877780 

9-9391 63;  10-060837-60 

3 

9-8u8218 

9-884148 

9-924070 

10-0759301 

9-817088 

9 877670 

9-93941 8!  10-060582150 

2 

9-808368 

9-884042 

9-924327 

10-075673 

9-817233 

9877560 

9-9397o3il0-060527 

58 

9808519 

9-883936 

9*924583 

10-075417 

9-817379 

9-877450 

9-9399281 10-060072 

57 

4 

9-803669 

9-883829 

9*yii-t840 

10-075160 

9-817524 

9-877340 

9-940 1831 10-059817 

5 

9-80S819 

9-88372 3 

9-925096 

10-074904 

1 9-817668 

9-877230 

9-94u439  i >059561 

0 

9-808969 

9-883617 

9 925352 

10-074648 

9-817813 

9-87712U 

9-940694' 10-U59300 

54 

r 

9-809  1 9 

9-883510 

9-325609 

10-074391 

9-817958 

9-877010 

9-940949' 10-05905! 

53 

s 

9-809269 

9-883404 

9-925865 

10  074135 

9-818108 

9-876899 

9-941204  10-05879- 

52 

S 9 

9-809419 

9-883297 

9-9261-22 

10-073878 

9 818247 

9-876789 

9-941459. 10-058541 

;i 

5 10 

9-809569 

9-8831*1 

9-926378 

10-073622 

9-818392 

9-876678 

9-94 1 71 3;  10-058- S7 

50 

9-809718 

9-883084 

9-926634 

10-073366 

9-818536 

9*876568 

9-941968  10-05805-. 

|49 

i 

9-809868 

9-882977 

9-926890 

10-073110 

9-818681 

9-876457 

9-942223!  10-057777 

48 

1 13 

9-S10017 

9-882871 

9-927147 

10-072853 

9-818825 

9-S76347 

9-942478  10-057522 

47 

9-810167 

9-882764 

9-927403 

10-072597 

9-818969 

9-876236 

9-9427.33  10-057267 

46 

1 15 

9-810316 

9-882657 

9-927659 

10-072341 

9-819U3 

9-876125 

9-942988  10-057012 

45 

9 ’Si 0465 

9-882550 

91927915 

10072085 

9-819257 

9-876014 

9-9432431 10-056757 

*44 

1 <6 

9-810614 

9-882443 

9-928171 

10-071829 

9-819401 

9-875904 

9-943498|  1 0-056502 

43 

1 is 

9-310765 

9-882336 

9-928427 

10-071573 

9-819545 

9 875793 

9-943752,10-056248 

42 

1 1 9 

9-810912 

9-882229 

9-92S684 

10-071316 

9-819689 

9-875682 

9-944007’ 10-055993 

41 

9-811061 

9-882121 

9-928940 

10-071060 

9-819832 

9*875571 

9-944262  10-055738 

40 

St71 

9-811210 

9-882014 

9-929196 

10-070804 

9-819976 

9-87545y 

9*94451/  10*055483 

39 

0 99 

9-811358 

9-SS1907 

9-929452 

10-070548 

9-820120 

9-875348 

9-944771  10-05522- 

38 

1 23 

9*8 1 1507 

9-881799 

’9-929708 

10-070292 

9-820263 

9-S75237 

9-945026  10-054974 

Oi 

9-811655 

9-881692 

9-329964 

1U-070036 

9-820406 

9-875126 

9-945281  10-054719 

36 

1 25 

9-811804 

9-881554 

9-930220 

10-069780 

9-820550 

9-875014 

9*945535  10*054465 

35 

826 

9-811952 

9-881477 

9-930475 

10-069525 

G-820G93 

9-874903 

9-945790  10-054210 

54 

9-812100 

9-881369 

9-930731 

10069269 

9-820836 

9-874791 

9*946044  10*053955 

33 

1 28 

9-812248 

9-881261 

9-930987 

10-069013 

9-820979 

9-S746SU 

9-946299  10-053701 

52 

129 

9-812396 

9-881153 

9-931243 

10068757 

9-821122 

9-874568 

9*946554  10*053446 

31 

9-812544 

9-S81U4G 

9-931499 

10-068501 

9-821265 

9-874456 

9-946808,10-053192 

.50 

31 

9-812692 

9-880938 

9-931755 

10  068245 

9-S21407 

9-874344 

9-947063  10-052937 

29 

[32 

9-812840 

9-880830 

9-932010 

10  067990 

9-821550 

9-874232 

9-947318  10052682 

28 

9-312988 

9-S80722 

9-932266 

10-067734 

9-821693 

9-874121 

9-947572  10-052428 

-27 

34 

9-813135 

9-880613 

9-932522 

10-067478 

9-821835 

9-874009 

9-947827  10052173 

26 

9-813283 

9-880505 

9-932778 

10067222 

9-821977 

9-873896 

9-948081  10-0519i9|25 

36 

9-813430 

9-880397 

9-933053 

10-066967 

9-822120 

9-873784 

9-948335 110051 665  (24 

37 

9-813578 

9-880289 

9-9S3289 

10-066711 

9-82226 -2 

9-S73672 

9-948590*  10051 4 HK2S 

38 

9-813725 

9-S801S0 

9-933545 

10-066455 

9-822404 

9-873560 

9-94884410051156:22 

j 39 

9-813872 

9-880072 

9-933800 

10-066200 

9-822546 

9-873448 

9-949099  10050901 '21 

Uo 

9-8 14019 

9-879963 

9-934056 

10065944 

9-82268S 

9*873535 

9-949353,10050647 

20 

9-814166 

9-879855 

9*9o43 1 1 

10065689 

9-822830 

9-873223 

9-94960S  10-050392  19 

42 

9-814313 

9-879746 

9*934567 

10-065433 

9-822972 

9-873110 

9-949S62  10050138  IS 

43 

9 S 14460 

9-879637 

9-934822 

1O-O0517S 

9-823114 

9-S7299S 

9-950116  10-049884!  17 

9 -8 14607 

9-879529 

9-935078 

10-064922 

9-823255 

9-872885 

9-950371  10049629116 

9-814753 

9-879420 

9-935333 

10-064667 

9-823397 

9-872772 

9-950625  10049376-15 

9-814900 

9-879311 

9*935589 

10-064411 

9-823539 

9-872659 

9-950S79  10-049121!  14 

47 

9-815046 

9-87.202 

9-935844 

10-064156 

9-823680 

9-872547 

9-951133  10  048867 

13 

248 

9-815193 

9-879093 

9-936100 

10-063900 

9-823821 

9-872434 

9-9513SS  10-048612  12 

iQ 

q -81 5339 

9-878984 

9-936355 

10-063645 

9-823963 

9-872321 

9-951642,10048358 

11 

50 

9-815485 

9-878S75 

9-9.36611 

10-063389 

9-824104 

9-S72208 

9-951 896  10-048104' 10 

9-815632 

9-878766 

9-936866 

10  063134 

9-824245 

9-872095 

9-95215010047850 

9 

j 

9-815778 

9-878656 

9-9.37121 

IO-062S79 

9-824386 

9-S71981 

9-952405  10  047595 

8 

53 

9-815924 

P-S78547 

9-937377 

10-062023 

9-824527 

9-871868 

9-952659  10-047341 

7 

54 

9-816069 

9-87843S 

9-937632 

10-062368 

9-824668 

9-S71755 

9-952913. 10-0470S7 

6 

9-816215 

9-87832S 

9-937887 

10-062113 

9 824808 

9-S71641 

9-953167, 10O46S33 

5 

56 

9-816361 

9-878219 

9-938142 

10-061858 

9-824949 

9-871528 

9-953421 1 10045579 

4 

57 

9-816507 

9-878109 

9-938398 

10-061602 

9-S25090 

9-S71414 

9-953675  10-046325 

3 

58 

9-816652 

9-877999 

9-938653 

10-061347 

9-825230 

9-87130! 

9-95S929!  10046071 

@59 

9-816798 

9-877890 

9-938908 

10-061092 

9-825371 

9-871187 

9-95418.5,10-045817 

1 

§6° 

9-816943 

9-877780 

9-939163 

10-060837 

9-S25511 

9-871073 

9-954437' 10-045565 

0 

1 

Cosine. 

Sine. 

Cotan. 

Tang. 

Cosine. 

Sine. 

Cotan.  I I ang. 

4S  Deg. 


LOG.  SINES,  TANGENTS,  be. 


i 'I  i ■ HI  I1  BH  I I i I II  l II  iTill  iilTTI  I I HPB  ii  I i iT'IT 

42L)eg.  43  Deg. 


'I 

Sine. 

Cosine. 

Tang. 

Cotang. 

Sine. 

Cosine.  | 

Tang.  | 

Cotang. 

50 

{) 

9-825511 

9-871073 

9-954437 

0*045563 

9-S337S3 

9-864127 

9-969656,10-030344 

1 

9-825651 

9-87' i960 

9-95469! 

10-045309 

9-833919 

9-864010 

9-96990P] 

0-030091 

l{) 

2 

9-825791 

9-870846 

9-954940 

10-045054 

9-834054 

9-S63892 

9-970162|  10-029338 

58 

3 

9-825931 

9-870732 

9-955200 

10-044800 

9-834189 

9-863774 

9-97041 6,10-02958  i 

57  i 

4- 

9-826071 

9-87061S 

9*955454 

10-044546 

9-334325 

9-S63656 

9-970660  10-02933! 

56 

5 

9-826211 

9-870504 

9-955708 

10-044292 

9*834460 

9-863538 

9-970922!  10-029078 

55 

r 

9-826351 

9-870390 

9-955961 

10-044039 

9-834595 

9-86341 J 

9-971 175. 19-028825 

54 

7 

9-826491 

9-870276 

9-956215 

10-043785 

9-834730 

9-863301 

9-971429;  10-028571 

5S 

8 

9-826631 

9-870161 

9-956469 

10-043531 

9-834865 

9-863183 

9-97 1682!  10-028318 

52 

e 

9-826770 

9-870047 

9-956723 

10-043277 

9-834999 

9-863064 

9-971935  10028065 

51 

l(i 

9-826910 

9-869933 

9-956977 

L0*043U23 

9-835134 

9-862946 

9-972188:10-027812 

10 

a 

9-827049 

9-869818 

9-957231 

10-042769 

' 9-835269 

9-864827 

9-972441110-027559 

V9 

12 

9-827189 

9-869704 

9-957485 

10-042515 

9-835403 

9-862709 

9-972695|  10-027305 

4S 

13 

9-S27329 

9-869589 

9-957739 

10-042261 

9-835538 

9-862590 

9-972948  10-027052 

47 

14 

9-827467 

9-869474 

9-957993 

10-042007 

9*835672 

9-862471 

9-975201 1 10-026799 

46 

9-827606 

9-S69360 

9-958247 

10-041753 

9-835807 

9-862353 

9-973454110-026546 

45 

16 

9-827745 

9-869245 

9-958500 

10-041500 

9-83594 

9-86-2234 

9-973707,10-025293 

44 

17 

9-827884 

9-869130 

9-958754 

10-041246 

9-836075 

9 862115 

9-9739601 10-020040 

43 

IS 

9-828023 

9-869015 

9-959008 

10-040992 

9-836209 

9-861996 

9-974213 

10-025787 

42 

19 

9 828162 

9-868900 

9-959262 

10-040738 

9-836343 

9-861877 

9-974466  10-025534 

41 

20 

9-828301 

9-868785 

9-959516 

tO'040484 

9-836477 

9-861758 

9-974720i  10-0252S0 

40 

21 

9-828439 

9-888670 

9-959769 

10-040231 

9-836611 

9-861638 

9-974973!  10-025027 

39 

22 

9-828578 

9*868555 

9-960023 

10-039977 

9-836745 

9-861519 

9-9752261 10-024774 

38 

23 

9-828716 

9-868440 

9-960277 

10-039723 

9-836878 

9-861400 

9-975479!  10-024521 

37 

24 

9-828855 

9-868324 

9-960530 

10-039470 

9-837012 

9-861280 

9-975732. 10-024268 

36 

25 

9-828993 

9-868209 

9-960784 

10-039216 

9-837146 

9-861161 

9-9759851 10.024015 

35 

26 

9-829131 

9-868093 

9-961038 

10-038962 

9-837279 

9-861041 

9-976238110-023762 

34 

27 

9-829269 

9-867978 

9-961292 

10-038708 

9-837412 

9-860922 

9-976491 110-023509 

33 

28 

9-S29407 

9-867862 

9-961545 

10-038455 

9-837546 

9-860802 

9-976744110.023256 

32 

29 

9-829545 

9-867747 

9-961799 

10-038201 

9-837679 

9-86J682 

9-976997 

10-023003 

31 

30 

9-829683 

9-867631 

9-962052 

10-037948 

9-837812 

9-860562 

9-977250 

10-022750 

30 

31 

9-829821 

9-867515 

9-962306 

10-037694 

9-837945 

9-860442 

9-977503 

10-022497 

29 

32 

9-829959 

9-867399 

9-962560 

10-037440 

9-838078 

9-860322 

9-977756 

10-022244 

28 

33 

9-830097 

9-867283 

9-962813 

10-0371S7 

9-838211 

9-850202 

9-978009 

10-021991 

27 

34 

9-830234 

9-867167 

9-963067 

10-036933 

9-838344 

9-860082 

9-978262 

iu-021738 

26 

35 

9-830372 

9-867051 

9-963320 

10-036680 

9*838477 

9-S59962 

9*978515 

10-021485 

25 

36 

9-830509 

9-866935 

9-963574 

10-036426 

9-838610 

9-659842 

9-978768 

10-021232 

24 

37 

9-830646 

9-866819 

9-963828 

10*036172 

9-838742 

9-859721 

9 979021 

10-020979 

2.3 

38 

9-830784 

9-866703 

9-964081 

10-035919 

9-838875 

9-859601 

9-979274 

10-020726 

22 

39 

9-830921 

9-866586 

9-964335 

10-035665 

9-839007 

9-859480 

9-979527 

10-020473 

21 

40 

9-831058 

9-866470 

9 964588 

10-035412 

9-839140 

9-859360 

9-979780 

10-020220 

20 

41 

9-831195 

9-866353 

9-964842 

10035158 

9-839272 

9-859239 

9-980033 

10-019967 

19 

42 

9-831332 

9-866237 

9-965095 

10  034905 

9-839404 

9-859119 

9-980286 

10-019714 

18 

43 

9-831469 

9-866120 

9-965349 

10-034651 

9-839536 

9-858998 

9-980538 

10-019462 

17 

44 

9-831606 

9-866004 

9-965602 

10-034398 

9-839668 

9-858877 

9-980791 

10-019209 

16 

45 

9-831742 

9-S65887 

9-965855 

10034145 

9-839800 

9-858750 

9-981044 

10-018956 

15 

46 

9-831879 

9-86577C 

9-966109 

10-033891 

9-839932 

9-858635 

9-981297  10-018703 

14 

47 

9-832015 

9-865652 

9-966362 

10-03363S 

9-840064 

9-858514 

9-981550:10-018450 

13 

48 

9-832152 

9-865536 

9-966616 

10-033384 

9-840196 

9 858393 

9-981803  10-018197 

12 

49 

9-832288 

9-865419 

9-966869 

10-033131 

9-840328 

9-858272 

9-982056' 10-01 7944 

11 

50 

9-832425 

9-865302 

9-967122 

10-032877 

9-840459 

9-858151 

9-9823091 10  017691 

10 

51 

9-832561 

9-865185 

9-967376 

10-032624 

9-840591 

9 858029 

9-9825621 10-01 7438 

9 

52 

9-832697 

9-86506S 

9 967621 

10-0.32371 

9-840722 

9-85790S 

9-982814 

10-017186 

8 

53 

9-832833 

9-864951 

9-967882 

10-032117 

9-840S54 

9-857786 

9-983067 

10-016933 

7 

54 

9-832969 

9-864832 

9-968136 

10-031864 

9-840985 

9-857665 

9-983320 

10-016680 

6 

55 

9-833105 

9-86471 

9-968389 

10-031611 

9-841116 

9-857543 

9-983573  10-016427 

5 

56 

9-833241 

9-864598 

9-968642 

10-031257 

9-841247 

9-857422 

9-983826  10-016174 

4 

57 

9-833377 

9-864481 

9-968S96 

10-031104 

9"84'37S 

9-857300'  g-9,84079  10-015921 

3 

58 

9*833512 

9-864362 

9-969149 

10-030851 

9-841509 

9-857178 

9-984332  10-015668 

2 

59 

9-833648 

9-864245 

9-969402 

16-030597 

9-84184C 

9-857056 

9-98458410  015416 

1 

60 , 9-833783 

9-864127 

9-969656 

10-030344 

9-841771 

9-856934 

9-984837!  10-01516. 

0 

Cosine 

Sine. 

C.otan. 

Tang. 

Cosine. 

Sine. 

Cotan. 

' Tang. 

' 

47  Deg. 


46 


log.  srvss,  i'WGGxrs,  &c. 

44  Deg. 


41. 


^me. 
9-84!  771 
9-84!  902| 
9-842033 
9-842163 
9-842294 
9-842424 
6 9-842555 
9-842685 
9-842815 
9-842940 
9-843076 
9-843200 
9*843336 


9-8434fifi 

9-843595 

9-843725 

9-843855 


9-843984  9-854850; 
9-844114  9-8547271 


13 

14 

15 

16 

17 

18 

19  9-844243 

20  9-844372 
9-844502 
9*844631 
9-844760 
9-844889 


Cosine. 

9-8569.34 

9-856812 

9-856610 

9-856=68 

9-856446 

9-856323 

9-850201 

9-856078 

9-855956 

9-855833 

9-855711 

9-855588 

9-855465 

9-855342 

9-852519 

9-8550961 

9-854973! 


I'an^ 


-984857 
985090 
•985343 
-985596 
9858481 
986101 J 
986354 


| Cotari; 

1O-015I63 

10-014910 

10-014657 

10-014404 

10-014152 

10-013899 

10-013646 


60 

59 

58 

57 

56 

55 

54 


9-986607!  10-013393  53 
9-986860;  10-01 31 40|52 

9-987112  10-012888151 


9-987565 

9-987618 

9-987871 


10’012635|50 

10-012382  49 
10-012129148 


9-988128  10-011877  47 
9-988376  4 0-01 1624j46 
9-988629  100113711=5 
9-988882)10011 1 1 S 44 1 


9845018 

9-845147 

9-845276 

9-845405 

9-845533 

9-845662 


9-989134  19  010866  43 
9*1 8938 7 1 1 001 06 1 3 42 
9-854603,'  9-989640  10010360  41 
9-854480  9-989893  If,-010I07  40 
9-8543561  9-990145  '0-009855  59 
9-854233  9-990398  10-009602  38 
9-8541091  9-990651  10-00934937 
9-853986  9-990903  10-009097|36 
10-00884435 

10-008591  34 

10-008338-35 
10-008086  32 
10-007833,31 


9-853862 
9-853738 
9-853614 
9-853490 

9-853366  - 

9-853242!  9-992420  10-007580  30 


9-991156 
9-991400  : 
9-991662  1 
9-991914  1 
9-992167  i 


31 

9-845790 

9-8531  IS 

9-992672  10-007328  29 

32 

9-84591. 

9-852994 

9-992925  10-007075,28 

33 

9-846047 

9-852869 

9-993178  10-006822  27 

34 

9-846175 

9-852745 

9-993431  10  006569|26 

o5 

9-846304 

9-852620 

9-993683  10-0063)7 

25 

36 

9-840432 

9-852496 

9-993936 10-006064 

24 

S7 

9-846560 

9-852371 

9-994-180  10-005811 

23 

38 

9-846688 

9-852247 

9-994441 -10-005559 

22 

39 

9-846816 

9-852122 

9-994694!  10-005306 

21 

Vo 

9-846944 

9-851997 

9-994947!l0-005O53 

20 

41 

9-847071 

9-851872 

9-9951 99;  10-004801 

19 

42 

9-847 : 99 

9-851747 

9;995i52 

10-004548 

18 

13 

9-847327 

9-851622 

9-995705 

10-004295 

17 

44 

9-8474s4 

9-851497 

9 995957 

10-004043 

16 

45 

9 847582 

9-851372 

9-996210 

10-003790 

15 

43 

9-847709 

9-851246 

9-996463 

10-003537 

14 

4-r 

9-847836 

9-851121 

9-990715 

10-003285 

13 

4S 

9-847964 

9-850996 

9-996968 

10-003032 

12 

49 

9-848091 

9-850870 

9-997221 

10-002779 

11 

50 

9-848218 

9-850745 

9-997473 

10-002527 

10 

51 

9-848345 

9-850619 

9-997726 

10-002274) 

9 

52, 

9-848472 

9850493! 

9 997979 

10-002021  f 

8 

53} 

9-848599 

9-850368 

9-998231 

10-001769! 

7 

54; 

9'84S726j 

9-850242- 

9-9984  84j  10-001 51 6, 

6 

55 

9-848852' 

9-8501161 

9-998737  10  00126S; 

5 

56 

9-848979 

9 8+9990 

9-99S989  10-001 01  ll 

4 

57 

9-849106, 

9-849864 

9-999242  10  000758 

3 

581 

9-8492321 

9 849738 

9-999495  10-0O0505 

q 

59 

9-849359 

9-S496U 

9-999747  10-00025S 

i 

60 

9-849485 

9-849485! 

10-000000  10-000000 

0 

1 Cosine.  ! 

Sme.  I 

Cotan. 

T ang. 

.■ 

45  Deg. 


( 


Date  Due 

( 

L.  B.  Cat.  No.  1137 

510 

Hutton 

H984  ? V2  23471 

Hat Hematics 

DATE  DUE 

ISSUED  TO 

0 


H9S4  P V;2  23471 


